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SANDIA REPORT SAND2012-8105 Unlimited Release Printed September 2012 Ultrashort Pulse Laser-Triggering of Long Gap High Voltage Switches Patrick Rambo, Jens Schwarz, Mark Kimmel, Aaron Van Tassle, Junji Urayama, Dale Welch, David Rose, Carsten Thoma, Robert E. Clark, Craig Miller, William R. Zimmerman, Todd A. Pitts, Mark R. Laine, Brenna Hautzenroeder, and Briggs Atherton Prepared by Sandia National Laboratories Albuquerque, New Mexico 87185 and Livermore, California 94550 Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energys National Nuclear Security Administration under contract DE-AC04-94AL85000. Approved for public release; further dissemination unlimited.

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Page 1: Ultrashort Pulse Laser-Triggering of Long Gap High …prod.sandia.gov/techlib/access-control.cgi/2012/128105.pdf · Ultrashort Pulse Laser-Triggering of Long Gap High Voltage Switches

SANDIA REPORTSAND2012-8105Unlimited ReleasePrinted September 2012

Ultrashort Pulse Laser-Triggering of LongGap High Voltage Switches

Patrick Rambo, Jens Schwarz, Mark Kimmel, Aaron Van Tassle, Junji Urayama, Dale Welch,David Rose, Carsten Thoma, Robert E. Clark, Craig Miller, William R. Zimmerman, Todd A.Pitts, Mark R. Laine, Brenna Hautzenroeder, and Briggs Atherton

Prepared bySandia National LaboratoriesAlbuquerque, New Mexico 87185 and Livermore, California 94550

Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation,a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of EnergysNational Nuclear Security Administration under contract DE-AC04-94AL85000.

Approved for public release; further dissemination unlimited.

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Issued by Sandia National Laboratories, operated for the United States Department of Energyby Sandia Corporation.

NOTICE: This report was prepared as an account of work sponsored by an agency of the UnitedStates Government. Neither the United States Government, nor any agency thereof, nor anyof their employees, nor any of their contractors, subcontractors, or their employees, make anywarranty, express or implied, or assume any legal liability or responsibility for the accuracy,completeness, or usefulness of any information, apparatus, product, or process disclosed, or rep-resent that its use would not infringe privately owned rights. Reference herein to any specificcommercial product, process, or service by trade name, trademark, manufacturer, or otherwise,does not necessarily constitute or imply its endorsement, recommendation, or favoring by theUnited States Government, any agency thereof, or any of their contractors or subcontractors.The views and opinions expressed herein do not necessarily state or reflect those of the UnitedStates Government, any agency thereof, or any of their contractors.

Printed in the United States of America. This report has been reproduced directly from the bestavailable copy.

Available to DOE and DOE contractors fromU.S. Department of EnergyOffice of Scientific and Technical InformationP.O. Box 62Oak Ridge, TN 37831

Telephone: (865) 576-8401Facsimile: (865) 576-5728E-Mail: [email protected] ordering: http://www.osti.gov/bridge

Available to the public fromU.S. Department of CommerceNational Technical Information Service5285 Port Royal RdSpringfield, VA 22161

Telephone: (800) 553-6847Facsimile: (703) 605-6900E-Mail: [email protected] ordering: http://www.ntis.gov/help/ordermethods.asp?loc=7-4-0#online

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SAND2012-8105Unlimited Release

Printed September 2012

Ultrashort Pulse Laser-Triggering of Long Gap HighVoltage Switches

Patrick Rambo, Jens Schwarz, Mark Kimmel, Aaron Van Tassle,Junji Urayama, Dale Welch∗, David Rose∗, Carsten Thoma∗,

Robert E. Clark∗, Craig Miller∗, William R. Zimmerman ∗, Todd A. Pitts,Mark R. Laine, Brenna Hautzenroeder, and Briggs Atherton

Abstract

Laser-triggered high voltage switches need longer length and lifetime laser plasmas atcontrolled plasma densities in order to achieve optimal operational parameters (such as shortruntime, low temporal jitter, and high voltage hold-off). We have approached the problemby considering ultrashort laser pulses of sub-picosecond duration for the high voltage triggersource. To this end, several laser plasma diagnostic techniques were employed to quantify theionization and to determine methods for appropriately tailoring the laser plasma for switch-ing purposes. Switching behavior under basic plasma conditions and with improved plasmachannels are discussed.

∗Voss Scientific LLC, 418 Washington Street SE, Albuquerque, NM 87108; www.vosssci.com; (505) 255-4201

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ContentsNomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Plasma Diagnostics and Ionization Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1 Optical Probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Wavefront Sensor Plasma Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3 Plasma Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Tailored and Enhanced Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1 Plasma Lengthening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Plasma Lifetime Increases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3 Plasma Density Increases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Laser-Triggered Discharges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.1 The Laser System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2 The High Voltage System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.3 Nanosecond Pulse Laser-Triggered Discharges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5 Nonlinear Electromagnetic Propagation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.1 Mathematical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.3 Validation and Evaluation of Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6 Laser-Triggered Switch Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.1 Particle-in-Cell Monte Carlo model for kinetic Simulations of laser triggering . . . . 806.2 Triggering with variation of laser parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.3 Integrated Simulation of Trigger Section of ZR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846.4 Discussion and Conclusions about the PIC Model of a Z LTGS . . . . . . . . . . . . . . . . 86

7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

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Figures1 The laser-triggered gas switch in the Z-Accelerator . . . . . . . . . . . . . . . . . . . . . . . . . 122 The laser ionization diagnostic layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Interferometry and subsequent asymmetric Abel inversion . . . . . . . . . . . . . . . . . . . . 204 Basic shadowgraphy of a laser plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Shadowgraphy of USP laser plasmas at varying energies . . . . . . . . . . . . . . . . . . . . . 236 Setup to measure the self-focus shift using a wavefront sensor (WFS) . . . . . . . . . . . 247 WFS examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 Plot of focal shift ∆ versus laser input energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Ionization rate versus intensity for oxygen and nitrogen . . . . . . . . . . . . . . . . . . . . . . 2810 Temporal detection of laser-generated plasma fluorescence in ambient air . . . . . . . . 3011 The effect of astigmatism upon plasma length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312 Plasma Florescence imaging of ns pulse laser-generated plasma concatenation . . . . 3413 USP laser-generated plasma concatenation based upon chromatic aberration . . . . . 3514 USP laser-generated plasma concatenation based upon self-focusing . . . . . . . . . . . . 3615 Plasma lifetime enhancement by double USP pulses with variable delay . . . . . . . . . 3716 Dual pulse USP plus nanosecond laser plasma generation . . . . . . . . . . . . . . . . . . . . 3917 OPCPA pump laser performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4118 Overview of the new four stage OPCPA system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4219 Zemax model of the temporal compressor layout . . . . . . . . . . . . . . . . . . . . . . . . . . . 4320 High Voltage switch stand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4621 Plasma fluorescence imaging for laser plasmas and triggered discharges . . . . . . . . . 4822 Air switch runtimes for 532 nm at various lens tilts and voltages across the gap . . . 5023 Air switch performance for 532 nm at various lens tilts for 104 kV across the gap . . 5124 Plasma fluorescence imaging for UV laser plasma sand triggered discharge . . . . . . 5225 Diffraction and Dispersion Lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7226 Critical Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7327 Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7428 Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7529 Raman and Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7630 Raman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7731 Raman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7732 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7833 Idealized representation of the LTGS trigger section used for the 2D simulations . . 8134 Electron density contour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8235 Simulations of the electric field and electron density in the trigger gap . . . . . . . . . . 8336 The number of excited SF6 atoms which relates directly to radiation production . . . 8437 A schematic of the connection of the simulation domain to the ZR circuit upstream

to the intermediate store and Marx and downstream through the PFL . . . . . . . . . . . 8538 Simulated PFL voltage plotted along with including data, circuit model and LSP

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

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Nomenclature

CCD Charge Coupled Device (camera)

CVR Current Viewing Resistor

DC Direct Current

FWHM Full Width at Half Maximum

HV High Voltage

IR InfraRed (light)

KDP Potassium Dihydrogen Phosphate

LIB Laser-Induced Breakdown

LSP Large-Scale Plasma

LTGS Laser-Triggered Gas Switch

MPI Multi-Photon Ionization

PIC Particle-In-Cell (code)

PMT Photo Multiplier Tube

PPT Perelomov, Popov, and Terentev

PSI Pounds per Square Inch

PSIA Pounds per Square Inch Absolute

PSIG Pounds per Square Inch Gauge

RMS Root Mean Square

USP Ultra-Short Pulse

UV Ultra-Violet (light)

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1 Introduction

Long-gap laser-triggered high voltage discharges have numerous applications from laser wakefieldacceleration [1] and super-radiant light sources (both based on extended controlled-density plas-mas) to laser-triggered spark gaps used for megavolt switching in pulsed power devices [2, 3] totriggered and guided lightning [4–10]. The problem facing all such efforts is the ability to createlong ionized channels with adjustable plasma density and extended plasma lifetimes in order toimprove discharge reliability, reduce temporal jitter, and allow for longer gaps and higher hold-offvoltages. To overcome these problems, we propose using ultrashort pulse (USP) sub-picosecondlasers to look at switching aspects in long-gap high voltage discharges, with special emphasis onextension of plasma lifetime. Application of ultrashort laser pulses (instead of the nanosecondpulses more commonly used in pulsed power switching) can create the desired extended lengthplasma channels via nonlinear filamentation processes [11] while having a very prompt plasmacreation time which can reduce switch jitters. Based upon these premises, existing work in thisarea has either focused on discharging the maximum length gaps with a USP laser (thus usingthe extended plasma channel length to minimize the ambient electric field for a given bias volt-age) [12–16] or focused on minimizing the jitter at very small gaps and low voltages [17, 18]. Atthe extremes, some USP laser-triggered discharge research has shown discharged gaps as large as3.8 m at about 2 MV [15] while other work has shown root-mean square (RMS) jitters as low as15 ps for 1 mm gaps at 4.5 kV [18]. Our goal is to pull together various aspects: lower jitters forlonger gaps and higher voltages.

To accomplish this, we must consider problems apparent in the existing literature. For example,the hindrance to increasing discharge lengths with USP lasers stems not only from the plasmalength but also the plasma lifetime and density. Filaments [11] occur when the USP laser powerexceeds the critical power Pcrit for self-focusing, which is a manifestation of the optical Kerr effectwherein the refractive index of the medium changes with intensity and causes the radial intensityprofile of the laser beam to establish a refractive index gradient and thus positively self-lens or self-focus the beam. The self-focusing continues until ionization (usually from multiphoton ionization)occurs significantly, cauing a negative lensing from the refractive index contribution of the plasmagradient. The two mechanisms balance in a process called intensity clamping, with the net resultthat multi-meter scale plasma channels at electron densities of around 1016 e−/cm3 can be produced(as compared to the number density of air at 1atm and 20C of 2.5x1019 molecules/cm3 [19]).Recombination and electron attachment reduce to the electron density in a filament to less than1013 e−/cm3 within a microsecond or so [13], with this density being considered a threshold levelto maintain when trying to guide a streamer [14]. A streamer is a self-maintaining ionizing pulsefront which moves along at 106 to 107 m/s while maintaining electron densities of∼1014 e−/cm3 atthe streamer tip [20]. Part of the barrier to longer gap discharges is the disappearance of electronsat one end of the gap before the streamer arrives. Put another way, the product of the streamervelocity and the electron lifetime to reach a threshold seed number density must equal or exceedthe target discharge gap length. To do this, either higher initial electron densities (which lead tofaster streamer velocities) or longer lifetime plasmas are necessary.

Since recombination and attachment processes increase with pressure, similar processes may

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come into play in a laser-triggered gas switch (LTGS) which operate at higher pressures (with 4 to10 atmospheres of either dry air or sulfur hexafluoride (SF6) being common to increase the hold-off voltage) over a shorter gap. In the case of the Z-Machine 6 MV LTGS [3] (see Figure 1), aconverging laser beam at 266 nm enters the switch gap area pressurized to around 70 PSIA. Thisdischarges roughly 1 MV across the trigger section, causing the subsequent cascade section to see avoltage change and associated increase in electric field. With the LTGS system operating at electricfields E and pressures P which constitute E/P values which are 80 to 85% of the value for self-breakdown of the gap, the voltage change associated with switching the trigger section throws thecascade section into self-breadown conditions. Subsequent self-breakdown discharges propagatedown the cascade electrodes to the cathode until the switch is fully conductive and the switch isclosed. However, this cascade section self-breakdown is more stochastic and thus inherently hasmore temporal jitter. If the length of the trigger section could increase with reliable functionality,then the cascade section length could decrease, thus decreasing the jitter. To do this, longer lengthlaser-generated plasmas are necessary in the trigger section.

Based upon these concepts, methods of measuring the plasma lifetime, plasma length, andplasma density will help with the optimization process. In addition, the ionization and recombi-nation dynamics are complex and are sometimes represented with conflicting formulas, constants,and explanations in the literature. This is in part due to the complexity of ionization processeswhich may occur individually or simultaneously, including:

1. Multiphoton ionization (MPI): A key process in filamentation wherein multiple photons aresimultaneously absorbed to bridge the ionization potenital of a given molecule. The processis dominant for lower gas pressures, shorter laser pulsewdiths, and shorter laser wavelengths.

2. Cascade ionization: Laser ionization mechanism wherein a free electron is accelerated byinverse Bremsstrahlung absorption of the laser light until the electron can impact ionizea neutral molecule, thus creating additional free electrons for the process to iterate upon.The process generally cascades until full ionization and high plasma temperature have beenachieved. The process is dominant for longer laser pulsewidths (> 1 ns), longer wavelengths,and higher gas pressures.

3. Tunneling ionization: For the most intense laser pulses, the electric field of the laser becomeslarge enough to distort the potential wells of the electron bound state in neutral molecules,enough so that the electrons can tunnel free through the well barrier.

To cite a specific problem, MPI is a key process in filamentation, with the process following amodel developed by Keldysh [21]. The original paper however cites multiple sample cases whichhave become distorted in subsequent less rigorous journal articles in the intervening years. Further-more, most of the literature indicates that MPI is pressure independent [11] while a few outlyingreferences cite a pressure dependence in the process [22]. Keldysh theory and subsequent variantslike the PPT theory (so-called because it was developed by Perelomov, Popov, and Terentev [23])have been benchmarked at very low pressures (≤ 10−7 Torr), but (to our knowledge) have not ap-propriately been validated at the higher pressures (≥103 Torr) that are present in an LTGS system.

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As such, the same plasma disgnostic methods used to tailor plasma parameters should be used toquantify the ionization processes involved.

With this in mind, we will first discuss plasma diagnostics and ionization measurements,followed by methods of tailoring the laser-produced plasmas, and finally laser-triggering of gasswitches. As modeling is an important part of any experiment, we wished to develop a simulationwhich could show switch operation from the laser plasma generation to the streamer developmentand subsequent high voltage breakdown. The USP laser ionization mechanism, as we will show,is full of complex simultaneous effects while the nansecond-scale laser ionization is a little moremanageable. In both cases, it is prudent to start LTGS simulation from a stand-point of the laserplasma providing a seed electron distribution on which the switch can work (rather than consider-ing the complex dynamics of simultaneous laser ionization and electron acceleration in an externalelectric field). To this end, one modeling effort focused on the laser-plasma generation methodswhile other focused on the HV switch. These efforts are discussed after the experimental data.

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trigger section

cascade section

trigger

electrodes

trigger gap

acrylic

insulator

cascade

electrodes

grading rings

anode

cathode

laser entry

SF6 gas-fill

Figure 1. The laser-triggered gas switch in the Z-Accelerator isfilled with 70 psia of SF6 and consists of a laser “trigger section”and a “cascade section”. The laser induced breakdown of the trig-ger section causes an over-voltage on the cascade section and startsthe self-breakdown process along the cascade electrodes. The pro-cess takes about 40-50 ns to complete.

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2 Plasma Diagnostics and Ionization Measurements

A systematic investigation should first examine ionization dynamics (including ionization by fila-mentation) through a suite of diagnostics techniques. The literature was examined for appropriatemethods of laser plasma detection, with key candidates being electron collection, optical probing(interferometry/shadowgraphy/Schlerein), spectroscopy, plasma fluroescence, and terahertz (THz)emission and probing. These are summarized here:

• Electron Collection: Electron collection uses the laser to create an ionized region betweentwo biased electrodes (either transverse [24–26] or longitudinal [26–29] to the laser beam)which subsequently collect the charge.

Timing Information: The method spatially integrates over the electrode spacing but canprovide temporal resolutions at the nanosecond scale depending on the level of charge inte-gration.

Spatial Information: Collection volume is integrated in the transverse dimension to thebias electric field. Longitudinal plasma dimension ambiguity is a problem for the transverseelectrode configuration due to edge effects but the length is better defined with longitudinalelectron collection. A simultaneous side-on image of the plasma length can also help. Col-lection vollume is also integrated along the bias electric field but, if the electrodes are smallrelative to the plasma length, then probing of the plasma length is possible.

Data Types: The technique yields the total number of collected charges. To get numberdensity, one must accurately know the detected plasma volume (which can be tricky). Lim-ited information about the plasma lifetime is available, depending upon the integration timesused.

Sensitivity: Electron collection can be very sensitive but is limited in precision (due toquestions about ionized volume and charge collection region) although the precision can beimproved if calibrated via a more precise external method. With calibration and higher biasvoltages, the technique can sense electron densities ≤ 1014 e−/cm3.

Nuances: Ideally, electrode separation and pressure should be small to avoid collisionaleffects during electron drift. Voltage should be low such that response is Ohmic and noavalanche breakdown is occurring. Protection of the electrodes with an insulator (used tomask the input laser beam) is necessary to avoid generating laser plasmas from the electrodesurfaces, thus contaminating the gas ionization results.

• Optical Probing: Optical probing typically involves othogonally delivering a synchronouslower power beam across the region of interest. This probing can be multichannel simulta-neously, meaning either multiple wavelengths can be used or beam splitting of the probe canallow multiple optical probing techniques to be used simultaneously [31].

Timing Information: With USP probe beams [30–33], precise temporal resolution (ofthe order of the probe pulsewidth) can be achieved. Temporal jitter is negligible if pump andprobe have a common source. Since the delay adjustment is optical and is thus based uponphysical pathlengths, spatial limitations usually restrict delays to ≤ 10 ns.

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Spatial Information: Transverse probing yields two-dimensional data which is spatiallyresolved (≥ 5 µm typically). Information is integrated along the direction of the probe beambut symmetry conditions may allows data disentanglement.

Data Types: Shadowgraphy can unfold absorption and refractive index (but the unfoldscan be difficult). Interferometry can unfold absorption and refractive index (and again theunfolds can be difficult but the literature is more abundant here for aid in analysis). Schleireindetects a change in refractive index (which does not in the end provide the data we seek).Abel inversion for cylindrically symmetric cases is a common unfold mechanism.

Sensitivity: Optical probing is limited in sensitivity to a dynamic range of 2 to 3 ordersof magnitude (1017 to 1019 e−/cm3) in part due to the ability to resolve fractional fringes (forinterferometry) and in part due to the bit depth of available cameras.

Nuances: All techniques do not distinguish between index changes due to electron den-sity and gas density changes (i.e. neutral particle gradients such as can be generated by ashock front). Gas density index change contribution is negligible for times below hydrody-namic expansion (≤1 ns)

• Spectroscopy: Plasma spectroscopic methods for USP laser-generated plasmas [34–36]collect the plasma self-emission and directs this light into a high resolution spectrometer foranalysis.

Timing Information: With a typical high resolution CCD as detector, spectroscopy tem-porally integrates the data. However, with a streak camera used to temporally resolve thespectrum (which will require significantly higher plasma emission and/or collection effi-ciency), picosecond-scale resolution can be achieved. Using a time-gated CCD at the spec-trometer output, resolutions of a few nanoseconds can be achieved. Similar temporal resolu-tion can be achieved if a photomultiplier tube (PMT) is used at the output and the spectrom-eter is used as a monochromator.

Spatial Information: Spectroscopy can be spatially resolved in one dimenion with animaging spectrometer or can be spatially integrated with input lenses for higher signal-to-noise. However, with the exception of a time-gated CCD on the spectrometer output, time-resolved methods typically integrate the spatial information.

Data Types: Spectroscopy can yield electron density information based on the broaden-ing of specific plasma emission lines [34,36] or electron temperatures based on the ratios ofthe amplitudes of specific plasma emission lines [34, 35].

Sensitivity: Spectroscopy for electron density measurements is limited in sensitivity toa dynamic range of 2 to 3 orders of magnitude (1017 to 1019 e−/cm3) in part due to limitedspectral resolution for common spectrometers and in part due to the bit depth of availablecameras. Collection efficiency, spectrometer efficiency, and plasma emissivity also limitsensitivity.

Nuances: Spectroscopy can indicate gas plasma chemistry (which is relevant for subse-quent laser pulses which interact with new species).

• Plasma Fluorescence: Plasma fluorescence methods [26,37] collect the plasma self-emission(as with spectroscopy) but use lower loss and lower cost filters in specific spectral regions to

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increase the signal strength. In addition , the detection method without a spectrometer canoffer different information about the plasmas.

Timing Information: In using a PMT [26], nanosecond timescale resolutions (althoughspatially integrated) are achieveable. Similarly, a time-gated image intensified CCD canoffer nanosecond timescale resolutions but with two dimensional spatially resolved imageframes.

Spatial Information: Unless using a time-gated image intensified CCD, a CCD withimaging collection optics will offer two dimensional spatially resolved but temporally inte-grated frames [37].

Data Types: Plasma lifetime and/or plasma density can be obtained in relative sense.However, when calbrated to an absolute measurement method, plasma fluorescence can pro-vide a sensitive absolute measurement of electron density or lifetime.

Sensitivity: When used in conjunction with a highly sensitive detection method such as aPMT or an intensified CCD camera, plasmas of 1016 to 1019 e−/cm3 can be detected. Whengains are turned up on such detectors and calibrated filters are used, the sensitivity can beincreased by several additional orders of magnitude to as low as 1013 e−/cm3 [37].

Nuances: Ideally, to provide reliable electron information and not just molecular excitedstate information, the plasma fluorescence can be used at known spectral bands corrspondingto the electron parameters. In air, the emission strength of the 391 nm spectral line associatedwith N2

+ has been shown to scale with electron density [38]. For high gain detectors suchas PMT’s or intensified CCD’s, detector linearity can be a concern.

• THz Emission and Probing: In the case of filaments, THz self-emssion has been observedand scales with electron density [63]. More generally, THz can be used like an optical probe(albeit with different techniques) with high temporal resolution [33,41,42]. Due to the muchlonger wavelength for THz than traditional optical probes, the level of plasma density towhich the probe is sensitive is much lower

Timing Information: The THz light used for probing as well as the filament emission areof picosecond scale of less.

Spatial Information: Spatial resolution at these long wavelengths is difficult with theplasma transverse dimension at 0.1 mm scale, which is small compared to the probe wave-length. However, if the plasma length is long, it can be resolved with 1 to 10 mm sacleresolution.

Data Types: The THz probing and emission can be used to determine electron density.

Sensitivity: Optical pump-THz probe methods are not usually single shot but can covera range of 1013 to 1017 e−/cm3 in a multishot mode [42]. This is consistent with singlecycle THz beams with frequencies from DC to around 3 THz. By comparison, filament THzself-emission correlates to electron density and has been calibrated with absolute methods toindicate ≥ 1015 e−/cm3.

Nuances: The generation and detection methods for THz are more elaborate than tradi-tional optical methods.

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Based upon complexity and possible safety issues, THz and electron collection methods weredeemed undesirable. This left optical probing, plasma fluorescence, and spectroscopy were ini-tally explored. Eventually, spectroscopy was dropped due to poor sensitivity since only the verybrightest of plasmas were detectable by a traditional 16 bit CCD behind the spectrometer. Withthese kinds of diagnostics in mind, an elaborate setup (see Figure 2) was developed which couldbe used for measurements of basic ionization information as well as measuring tailored plasmaspatiotemporal distributions.

1054nm 500fs 20mJ

532nm 8ns 150mJ

OPCPA

λ/2

P

P

∆t ∆t ∆t

LP1 BS LP2

MP1

LI1 LI2

LD1

LD2

EMfs

EMns

CFFfs

CFFns

PH

EMtrans

PD Gas

Ionization Cell

MP2

∆t

Probe Beam

Two-Color Interferometry/

Schlieren/ Shadography

PMT

CWFS

CP1w

CP2w

Figure 2. The laser ionization diagnostic layout. The USP laserreceives an initial split at a beamsplitter (BS) to allow a weakdelay-adjustable (∆ t) optical probe at 1054 nm and/or 527 nm to begenerated. A subsequent half waveplate (λ /2) and polarizer P al-low the generation of collinear delay-adjustable s and p-polarizedUSP pulses for laser ionization. A dichroic allows a collinearnanosecond-scale 532 nm laser for ionization in the gas cell. Leak-age at the dichroic is used to measure the intensity of the ionizinglaser pulses via cameras (C), energy meters (EM), and photodiodes(PD).

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2.1 Optical Probes

Interferometry

Interferometry was deemed to best initial diagnostic choice for its time-resolved two-dimensionalimages which could be directly unfolded to yield absolute electron density measurements. Inconjunction, leakage light of the ionizing laser pulse (as taken at the final fold dichroic optic beforethe ionization cell) is used to establish an equivalent image plane of the focal spot at much lowerintensities. The same leaked light would also provide the calibrated laser energy and pulsewidthsuch that the overall focal intensity in the ionization cell should be known. Knowing the focalintensity and the resultant electron density from interferometry should design the key aspects of theionization process. However, one should question the source of the measured plasma interferogramfringe shifts as well as the integrity of the equivalent focal intensity measurement in order tomaintain the validity of the final data analysis.

With respect to the interferometry, at early times, electrons are the main contributor to thisrefractive index data. However, at later times, hydrodynamic evolution of the plasma shock frontcreates a varying density of neutral particles which is convoluted with the electron density data.With ony one piece of data and two unknowns, an ambiguity is thus created in the interferometricdata between the effects of the neutral and electron densities. To deal with this, nanosecond pulsedlasers and continuous wave lasers have been used to provide simultaneous two-color interferome-try, providing two pieces of data for two unknowns [43–46].

With regards to the experimental setup, we have set up a femtosecond laser pump-probe exper-iment where one part of a 500 fs laser pulse with 20 mJ at 10 Hz at 1054 nm creates an atmosphericplasma in a test cell and a 10% leakage part of that pulse for transverse probing purposes. Theprobe beam is spatially filtered and delayed from -1 ns to >10 ns with respect to the main beam. Athin potassium dihydrogen phosphate (KDP) crystal in the probe beam allows frequency doublingto 527 nm such that the collimated fundamental (at 1 cm diameter beamsize) and second harmonicprobe the plasma with negligible delay. This delay is roughly 1 ps and consists of the sum of thedelays from the group velocity mismatch in the 3 mm thick KDP doubler and the 1 inch thickfused silica probe beam input window. After probing the plasma, the beam is enlarged by an allreflective beam expander which magnifies the probe region by 3 times for better resolution at asubsequent interferometer camera. This reflective telescope uses longer focal length (f1=50 cmand f2=150 cm) spherical curvature mirrors at shallow incidence angles to mitigate any aberrations(specifically any astigmatism resulting from the off-axis use). Protected silver coatings are used inthe telescope and in all subsequent high reflectors to support large bandwidths.

After the beam expansion, the beam goes to an interferometer. Initially, an easy approach wastaken via the simple air wedge lateral shearing interferometer [31, 47]. However, for a collimatedprobe beam input, this interferometer generates two images offset from each other by the wedgeangle. For long plasma lengths along the wedge direction, these images can blur together and createconvoluted interferograms. As such, a modified Michelson interferometer (similar to [30]) wasdesigned with a hollow roof prism made of silver mirrors in one arm and a flat silver high reflectorin the other. For an appropriately placed beam, the roof mirror will allow the test region of the

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probe beam to interfere with an unperturbed region of the same beam (i.e. a lateral displacementrather than a lateral shear interferometer). A beamsplitter and compensator plate are necessary inthe interferometer in order for the group delays for the two probe beam wavelengths to be the same.A dichroic is used after the interferometer to separate the two different wavelength interferograms.Different color and neutral density filters are used to provide unambiguous interferograms in eachvertical portion of the camera.

For a plasma being probed, the interferogram obtained provides total phase shift information∆φtot which is the sum of phase shift contributions from electrons (∆φe) and from neutral den-sity particles (∆φ0). Generally speaking, the low mass and high mobility of the laser-generatedelectrons will lead to early time phase data being dominated by the electron contribution while thehigher mass and lower mobility of the neutral particles will dominate at later times (especially withrecombination losses decreasing the electron contribution). The generalized phase shift ∆φ can beexpressed in terms of probe wavelength λ and the line-integrated medium refractive index n alongthe direction x over length L as:

∆φ =2π

λ

∫ L

0(n(x)−1)dx. (1)

For a plasma with electron density Ne, the refractive index ne due to electrons is expressed [49]:

ne =

√1− λ 2e2Ne

4π2ε0mec2 =

√1− Ne

Nc≈ 1− Ne

2Nc= 1− λ 2Nere

2π, (2)

where Nc = (4π2ε0mec2)/(λ 2e2) is the cutoff electron density and the approximation assumes thatthe refractive index ne∼ 1 such that (n2

e−1)= 2(ne−1). The wavelength dependence is broken outof the cut-off density by expressing it in terms of the classical electron radius re = e2/(4πε0mec2)[46] such that Nc = π/(λ 2re).

Similarly, for a neutral gas of density N0, the refractive index due to the neutral gas n0 is expressed[48] in terms of a modified Cauchy relation:

n0−1 =N0

Natm[Aatm · (1+

Batm

λ 2 )] =N0

Natmβ , (3)

where Natm is the number density at standard atmospheric pressure and Aatm and Batm are fittingparameters dependent on the gas type (as measured at standard atmospheric pressure) which canbe consolidated into the parameter β (which for air is the Gladstone-Dale constant [43]).

Based upon these refractive indices, the phase shift contributions from electrons (∆φe) and fromneutral density particles (∆φ0) are (respectively):

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∆φe =2π

λ

∫ L

0(ne(x)−1)dx =−λ re

∫ L

0Ne(x)dx, (4)

∆φ0 =2π

λ

∫ L

0(n0(x)−1)dx =

2πβ

λNatm

∫ L

0N0(x)dx. (5)

When interferograms are recorded at two wavelengths λ1 and λ2, equations 4 and 5 at these twowavelengths can be combined to yield:

∫ L

0Ne(x)dx =

λ1∆φλ1−λ2∆φλ2

re(λ 22 −λ 2

1 ), (6)

∫ L

0N0(x)dx =

Natm(λ2∆φλ1−λ1∆φλ2)

2πβ (λ2/λ1−λ1/λ2). (7)

With this derivation in mind, we record approximately synchronous two-color interferogramsfirst and then calculate the phases. To do this, we employ the standard spatial carrier method [50].This process takes the Fourier transform of the interferogram image, which yields a strong centerfrequency along with two symmetrically placed side-lobes at the frequency of the interferogramfringes (or the spatial carrier). One of these side-lobes is extracted and shifted back to the centerfrequency, A subsequent inverse Fourier transform is taken to yield the phase map. One shouldnote that, first of all, many fringes are desirable (at least more than 50 in practice) such that theinformation in frequency space of the central lobe does not spill over into the information from theside-lobes. Secondly, soft apodization filtering of the frequency side-lobe is necessary to mitigatespatial ringing in the phase image. Finally, in a similar manner, soft apodization filtering of theoverall image edges is necessary to mitigate ringing when going from either spatial or frequencyimages to its Fourier transform analog.

Since phase determination methods wrap the phase to lie within the range of -π to π radians,a phase unwrap routine was written to look at the phase discontinuities and reproduce the correctcontinuous phase. Once unwrapped, the phases at the two wavelengths are used to determine theline-integrated electron and neutral densities as denoted in the left-hand side of Eqs. 6 and 7. Todo this in a practical manner, a spatial fiducial is used in each interferogram frame to provideregistration for the accurate combination of these two phase maps. The resulting line-integratednumber densities incorporate both the value of the number densities and the spatial extent of theregion in question. If cylindrical symmetry is assumed, one can disentangle these values with anAbel inversion technique, yielding number density as a function of radial position. Researchersthat employ such methods may have extensive problems when plasma symmetries become clearlybroken in the line-integrated phase map. As such, an asymmetric Abel inversion technique (whichassumes a partial symmetry about the viewing plane) is used here [51,52]. For the two dimensionalphase map in the xz plane, this approach takes, about longitudinal beam propagation axis z at aposition x0, the phase maps ∆φplasma(x,z) in the upper half (from x ≥ x0) and lower half (fromx≤ x0) and relates them to a subsequebtly used integration variable η thus:

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Electron Density (1019 e-/cm3)

Plasma Phase Shift (Radians)

Position (m)

Raw Interferogram

Unwrapped Phase

Figure 3. Interferometry and subsequent asymmetric Abel inver-sion at a near zero delay.

∆φ+(η ,z) = ∆φplasma(x x0,z) for x > x0, (8)

∆φ(η ,z) = ∆φplasma(x0 x,z) for x < x0. (9)

These go into symmetrized and antisymmetrized half maps ∆φs and ∆φa respectively:

∆φs(η ,z) =12(∆φ

+(η ,z)+∆φ(η ,z)), (10)

∆φa(η ,z) =12(∆φ

+(η ,z)∆φ(η ,z)). (11)

Using these, two different two dimensional maps ne0(r,z) and ne1(r,z) are developed:

ne0(r,z) =ncλp

π2

∫∞

rdη

∂η[∆φs(η ,z)]√

η2 r2, (12)

ne1(r,z) =ncλpr

π2

∫∞

rdη

∂η[∆φa(η ,z)

η]√

η2 r2. (13)

(14)

The method assumes that the xz plane of the phase map consistutes a mirror symmetry plane

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such that the final three dimensional electron density ne is expressed:

ne(r,θ ,z) = ne0(r,z)+ne1(r,z)cos(θ) (15)

Note that, in the event that the data truly is symmetric about the z propagation axis, one gets∆φ+(η ,z) = ∆φ−(η ,z) = ∆φ(η ,z) which causes ∆φa(η ,z) = 0 and ∆φs(η ,z) = ∆φ(η ,z). At thatpoint, the form of ne(r,z) looks like the standard symmetric Abel inversion seen elsewhere [32].

We first used the technique on a single USP inferferogram, an example of which is shown inFigure 3. Using the process described, the unwrapped electron density (at θ = 0 in the xz plane)measures approximately 50% of full single ionization of all air molecules for a londitudinal posi-tion slightly away from the peak ionization. The process shows the applicability of the asymmetricAbel inversion technique, as this region would be difficult to force symmetry conditions upon in anormal Abel inversion approach. Subsequently, we tested the two-color interferogram method butfound negligible difference in the measured phases for the two color. This is expected because theUSP laser-generated plasma lifetime is only a few nanoseconds at best, meaning that the plasmadoes not have enough time to hydrodynamically evolve into a measureable neutral particle densitygradient while there is still a signficant electron density.

Shadowgraphy

In many cases, spatial information about the plasma extent may be adequate, by-passing the needfor rigorous interferometric analysis. Such data can be simply extracted by blocking the refer-ence arm of the inferometer to yield a shadowgraph. In this case, the optical probe sees localrefractive index and opacity differences from the plasma. Figure 4 compares shadowgraphs oflaser-generated plasmas from USP and nanosecond lasers pulses. Note that the nanosecond pulseplasma is extremely opaque and shows signs of beading plasma structure for a fairly significantlaser energy. By comparison, the USP plasma is longer and more continuous at a significantlylower laser energy. The shadowgraphs point out another important fact: Attempting to perform in-terferometry on nanosecond pulse laser-generated plasmas will not work due to the extremely hotdense plasma formed. If the shadowgraph is opaque in the plasma region, the interferogram willshow no fringe data in this region either. As such, pursuit of nanosecond laser plasma informationsubsequently focused upon plasma fluorescence and shadowpahy techniques.

A more detailed application of shadowgraphy was to examine the plasma length as a function ofUSP laser energy (see Figure 5). As the energy is turned up, two things occur: 1) The plasma length(as measured by the distribution FWHM) increases and 2) The plasma distribution (as measuredby the distribution centroid) extends back towards the focusing lens. This first point goes to theheart of creating tailored plasmas (see section 3). However, one should not be able to turn up theenergy infinitely and see the same plasma lengthening improvement. At a peak power of severaltimes the critical power Pcrit for self-focusing (which are energies unatainable with the system usedin this testing), the beam will fragment and begin to generate multiple filaments. The further thatthe peak laser power is above above Pcrit , the more filaments and associated plasma channels aregenerated. However, at the lower energies tested, as the plasma lengthens, it should be noted that

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1mm

1mm

Beam Direction

Beam Direction

(a)

(b)

Figure 4. Basic shadowgraphy of a laser plasma. In bothcases the laser enters from the left side and is focused by anf=25 cm lens in ambient air. The probe beam delay was setat the time of peak ionization (taken to be zero delay betweenpump and probe). The horizontal image length along is 5.5 mm.(a) USP laser (λ=1054 nm,φ=12 mm FWHM, 14 mJ, t∼500 fs)plasma. (b) Corresponding plasma for a nanosecond laser pulse(λ=532 nm,φ=8 mm FWHM, 300 mJ, t∼8 ns).

farthest distance of the plasma distribution is fairly consistent and remains near the farthest end ofthe lowest energy laser plasma created near the paraxial focus. As the plasma lengthens, it growsback towards the focusing lens, as one would expect in the presence of self-focusing. Thus, thelengthening of the plasma in one direction leads to a plasma centroid shift in that direction.

One should note that at the lowest energies, the plasma still seemed to be showing some cen-troid shift below Pcrit (although it was somewhat ambiguous). However, according to [54], theplasma should exhibit no further centroid shift at peak powers below Pcrit . The inabiliity to con-firm this in our experiment poses a problem in that the goal of the plasma diagnostics is to quantifyionization based on an equivalent intensity measurement to the ionizing laser pulse. If the fo-cal plane and focal spot size varies with energy, the equivalent intensity measurement is invalid.Further, elaboration of the low energy data in Figure 5 proved inconclusive because the evidenceof the plasma tends to vanish below 0.5 mJ for both shadowgraphs and interferograms. Abovethis energy, the effects of both self-focusing and plasma generation appear to be present. Thisissue prompted a more sensitive measurement technique, which is the time-integrated wavefrontmeasurement discussed in the following section.

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0.68mJ

0.93mJ

1.35mJ

3.57mJ

3.73mJ

6.53mJ

7.20mJ

8.73mJ

11.00mJ

1mm

0 2 4 6 8 10 120.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

channel length

chan

nel le

ngth

(mm

)

energy (mJ)

0.0

0.5

1.0

1.5

2.0

2.5

cent

roid

shi

ft (m

m)

centroid shift

P cr=4

GW

(2m

J/0.

5ps)

(a)

(b)

Figure 5. Shadowgraphy of a laser plasma. (a) The laser(λ=1054 nm,φ=12 mm FWHM, t∼500 fs) enters from the left side,as focused by an f=25 cm lens in ambient air. The arrow indicatesthe nominal paraxial focus. The probe beam delay was set at thetime of peak ionization (taken to be zero delay between pump andprobe). The horizontal image length along is 5.5 mm. (b) Imageanalysis of data from (a). Note that the critical power estimatedfrom [53] is shown to indicate where self-focusing should begin.

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2.2 Wavefront Sensor Plasma Diagnostics

New approach to measure MPI coefficients and n2

As shown in the previous section, for energies above 1 mJ, the measurement of MPI coefficientsand nonlinear index n2 cannot be performed since both phenomena are present simulatneously. Wepropose a new method for measuring n2 and ionization rates which allows us to detect beam shiftsat much lower energy (or power P Pcrit). In our setup (see Fig. 6), a wavefront sensor is usedto directly sense a shift in focal length as opposed to indirectly measuring shifts via a secondaryeffect such as plasma fluorescence [55]. This is critically important because plasma induced focalshifts occur well before plasma fluorescence can be detected.

R d

f2

M1

M2

WFS

D

f1

D

Figure 6. Setup to measure the self-focus shift using a wavefrontsensor (WFS). The blue lines represent the low peak power casewhile red lines represent powers sufficient for self-focusing.

A short pulse laser at low power P Pcrit is focused with a focal element M1 (focal lengthf 1) into a gas ionization cell and re-collimated with a second focal element M2 (focal length f 2).A wavefront sensor (WFS) is placed at a distance d behind M2 and a flat wavefront is recordedas a reference (see Fig. 7(a)) at minimum powers. Spherical mirrors were used in the focusingand re-collimation of the laser beam (see M1 and M2 in Fig. 6) in order to avoid nonlinear phaseaccrual in the lens material which may compromise data.

As the laser beam power increases, the beam focus will shift by a distance ∆ toward M1.Correspondingly, the light emanating from the new self-focused position f 1−∆ or f 2+∆ will notbe collimated any more but rather be imaged a certain distance D behind M2. This distance D isjust the radius of curvature R of the wavefront (as measured by the wavefront sensor; see Fig. 7(b))plus the distance d of the sensor to the lens M2: D = R+d. The focal shift ∆ can then be calculatedfrom the thin lens equation:

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(a) (b)

Figure 7. (a) Flat reference/initial wavefront recorded for thef/120 geometry at a beam energy of 0.3 mJ. The associated “de-focus” term was measured at R0

2 = −0.005 waves. (b) Wavefrontcurvature recorded at 5.4 mJ energy with R0

2 = 0.05 waves “defo-cus” for the same experimental configuration as (a).

1f 2

=1

f 2+∆+

1R+d

⇐⇒ ∆ =1

1f 2 −

1R+d

− f2. (16)

The wavefront radius R at position d can be calculated from the measured defocus Zernike termR0

2 (see Fig. 7(b)). In our case R02 = 2a2−1, where a is the pupil radius. For a measured pupil of

radius a on the wavefront sensor and a defocus of R02 in waves one calculates:

R =a2

4R02λ

, (17)

where λ is the wavelength of the laser.

Experimental results

A laser beam at λ = 1054 nm with a pulsewidth of τ = 540 fs FWHM and a spatial first orderGaussian with a 1/e2 waist radius of wi = 6.2 mm was focused by spherical concave mirrors off/120 and f/40. A larger f/# (greater than f/120) was used as well but led to the onset of filamen-tation at low laser energies. Each data point represents an average of 200 measurements at 10 Hzon the wavefront sensor (Phasics SID-4, see http://www.phasicscorp.com/) as well as the energymeter. The peak to valley “defocus” Zernike term R0

2 was then fitted to the averaged wavefront andconverted to focal shift as discussed earlier.

Figure 8 shows the measured focal shift versus laser input energy for the two f/# mentionedearlier. One can see that, depending on f/#, the onset (elbow in the graph) of the focal shift is

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0 . 0 0 0 0 . 0 0 2 0 . 0 0 4 0 . 0 0 6

0 . 0 0 0

0 . 0 0 5

0 . 0 1 0

0 . 0 1 5

0 . 0 2 0

0 . 0 2 5 d a t a f o r f / 4 0 d a t a f o r f / 1 2 0 f / 4 0 m o d e l w i t h n 2 a n d 1 1 t h o r d e r M P I f / 1 2 0 m o d e l w i t h n 2 a n d 1 1 t h o r d e r M P I f / 4 0 m o d e l n o M P I f / 1 2 0 m o d e l n o M P I

focal

shift

(m)

e n e r g y ( J )

0 2 4 6 8 1 0 1 2p o w e r ( G W )

Figure 8. Plot of focal shift ∆ versus laser input energy. Thescatter plots show experimental data for f/120 and f/40 focusinggeometries. The solid lines represent theoretical fits based on thepulse propagation equation 19 (see below) and the dashed linesdepict the theoretical behavior without ionization present.

different for any given focus geometry. Two main processes contribute to focal shift:

• Nonlinear self-focusing: As the laser beam power P becomes comparable to the criticalpower Pcrit = 3.77λ/(8πn0n2), the nonlinear index of refraction n2 leads to additional focus-ing in beams with Gaussian (and also other) intensity profiles. For a collimated input beamentering a nonlinear medium and an unchanged amplitude profile during self-focusing, onecan write the self-focusing distance zs f due to the nonlinear index n2 as [56]:

zs f =πw2

(P

Pcrit−1)−1/2

, (18)

where wi is the initial beam waist at the entrance of the nonlinear medium. In a focusinggeometry, the new effective focal length fnew can then be found as a combination of the ge-ometrical focal length fg through the lens transformation formula: fnew = fgzs f /( fg + zs f ).

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One can see that this will lead to a shift of the minimum beam waist (focus) toward the inputfocusing lens.

Based on Eq. 18, a focal shift can only occur at P≥ Pcr which would be independent of focusgeometry (see dashed lines in Fig. 8). Liu et al [54] uses this argument when he identifiesthis elbow as the onset of P = Pcr in order to calculate n2. According to measured data inFig. 8, the elbow is clearly dependent on beam focusing geometry and hence one cannotconsider the effects of nonlinear self-focusing alone.

• Ionization: It is clear that ionization must occur as the laser beam propagates towards theminimum beam waist. This ionization will be highest around the focal region and will pro-foundly affect the self-focusing behavior [57]. As the beam collapses towards the focus theelectron density increases as a result of ionization due to increased beam intensity. This leadsto plasma de-focusing before the geometrical focus is reached. Hence ionization will lead toa focal shift toward the input focal lens as will nonlinear self-focusing due to the nonlinearindex n2.

It can be seen that one has to consider both processes in order to adequately explain the resultspresented in Fig. 8. One should note that for f/40, where ionization is dominant, the focal shiftbegins at lower power but shows smaller shifts at larger powers as compared to the regime wherenonlinear self-focusing is dominant (f/120).

Experimental analysis

In order to extract values for both n2 and the ionization rates, one has to model the beam waistas a function of laser power P and laser induced ionization. We used the semianalytical modelemployed in [11, 57, 58] and it was shown that it adequately reproduces the beam waist evolutionuntil the minimum waist is reached [57] which is the regime of interest. The laser beam waistw(z, t) as a function of time and propagation distance z can be written as:

∂ 2w(z, t)∂ z2 =

4k2w(z, t)3

(1− P(t)

Pcrit

)+

2K(K +1)2n0Ncritw(z, t)

×Ne(z, t), (19)

where the first term describes diffraction and self-focusing and the last term describes the defo-cusing due to ionization. In this equation k is the wavenumber, P(t) is the laser power, Pcrit =3.77λ 2/8πn0n2 is the critical power for self-focusing for a Gaussian beam, and n2 is the nonlinearindex of the gas under study (air). K is the ionization order of the multi-photon ionization (MPI)process. For an ionization potential of WO2 = 12.06 eV for oxygen and a photon energy of 1.18 eVfor a photon at 1054 nm one calculates: K = d12.06eV/1.18eVe = 11. n0 is the index of air andNcrit = ε0meω2

l /e2 is the critical plasma density for the laser. Here, ε0 is the dielectric constant invacuum, me is the mass of the electron, ωl is the angular frequency of the laser, and e is the chargeof the electron.

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Since there exist two data sets and two unknowns (n2 and σ (K)), one should be able to finda unique solution that fits the measurement. This is accomplished rather easily since the physicsin the f/40 focusing geometry is largely dominated by ionization which sets the limit for σ (K).Conversely, self-focusing plays a dominant role in the f/120 focusing geometry which leads to adetermination of n2. The higher influence of ionization in the f/40 case is evidenced by the factthat the onset of a focal shift (elbow) occurs at much lower powers.

The importance of ionization for the correct estimation of n2 and σ (K) is illustrated by thedashed lines in Fig. 8. They are based on theoretical curves that do not include ionization. Asexpected, those curves are shifted to the right because the focal shift contribution from the laserplasma is omitted. If only one f/# measurement is considered (as is the case in [54]), one is inclinedto identify the elbow in the curve as Pcrit , leading to an erroneously low Pcrit (or conversely highn2).

1 0 1 5 1 0 1 6 1 0 1 7 1 0 1 8 1 0 1 9 1 0 2 01 0 - 1 61 0 - 1 41 0 - 1 21 0 - 1 01 0 - 81 0 - 61 0 - 41 0 - 21 0 01 0 21 0 41 0 61 0 8

1 0 1 01 0 1 21 0 1 41 0 1 61 0 1 8

ioniza

tion r

ate (1

/s)

i n t e n s i t y ( W / m 2 )

I o n i z a t i o n r a t e p r e d i c t e d b y w e a k f i e l d K e l d y s h a p p r o x i m a t i o n f o r O 2 a t 1 0 5 4 n m M e a s u r e d i o n i z a t i o n r a t e P P T m o d e l f o r o x y g e n a t 1 0 5 4 n m P P T m o d e l f o r o x y g e n a t 8 0 0 n m P P T m o d e l f o r n i t r o g e n a t 1 0 5 4 n m

S e e A p p e n d i x A f o r m o d e l p a r a m e t e r s

M P I a t γ > > 1 t u n n e l i n g i o n i z a t i o n a t γ < < 1

Figure 9. Ionization rate versus intensity for oxygen and nitro-gen. The dashed, vertical green line depicts the transition regionbetween MPI and tunneling ionization, where the Keldysh param-eter γ = 1 for λ = 1054 nm.

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The solid lines in Fig. 8 show the calculated focal shift versus energy for n2 = (2.6± 0.2)×10−23 m2/W and σ (11) = (3± 1.5)× 10−191m22W−11s−1. The value for n2 compares well withother data from the literature. On the other hand, one should be aware that the measured ioniza-tion coefficient for the 11th order process is 10 orders of magnitude lower than expected basedon a weak field approximation of the Keldysh theory. The reason for this discrepancy is due tothe fact that both focusing geometries reach an intensity clamped regime of Icl = 3.5×1018 W/m2

at which tunneling ionization dominates the ionization process. To illustrate this point, we havesimulated ionization rates for different gases using the model by Perelomov, Popov, and Terentev(PPT model) [23].

Figure 9 shows the calculated ionization rate R versus laser intensity for oxygen and nitrogen.Note, that dNe/dt = (NO2−Ne)×R. The model was benchmarked against ionization time of flightdata conducted with ultra short pulse lasers at pressures ≤ 10−7 Torr. One can see from Fig. 9 thatthe measured ionization rate at the clamped intensity agrees well with the PPT model, includingboth MPI and tunneling ionization.

Conclusions about the wavefront sensor method

We have presented a novel method for measuring the nonlinear index of gases and their corre-sponding ionization rates (in this case air/O2) under atmospheric pressure at λ = 1054 nm. It hasbeen pointed out that measured focal shifts result from a combination of self-focusing and plasma-defocusing which has been appropriately treated in the semi-analytical model. Based on thismodel, we have calculated effective MPI coefficients for O2 in the intensity regime of 1018 W/m2.The measured coefficient is in excellent agreement with the PPT model. This result has broaderimplications, because it allows one to safely use the presented PPT curves for modeling efforts onlaser propagation in air and other gases.

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2.3 Plasma Fluorescence

Plasma fluorescence methods can provide a powerful simple approach to plasma diagnostics. Thebroadband fluorescence will not provide information about electron density because the emissioncomes from a variety of pathways. However, filtered regions around 391 nm have been showncorrelate to the electron density [37, 38]. A slightly broader transmission region was achieved viaa combination of Schott BG39 and UG11 color filters (see Figure 10). In Figure 10a, the temporalduration of the plasma fluorescence is shown to increase with shorter f/# (short focal length) opticsfor the USP laser source. In the longer focal length case, the temporal pulsewidth correlates wellto the expected PMT response (which is expected since the ionizing pulse is much shorter thanthe PMT response time and the plasma lifetime for a relatively cold plasma is of similar or shortertimescale). In the case of the shorter focal length lens, the lifetime is significantly longer, indicatinga hotter and/or denser plasma with a longer recombination time. Based upon the anticpated 25times increase in intensity for f=5 cm lens compared to the f=25 cm one, this observation could bedue to an increase in tunneling ionization and/or additional heating from inverse Bremsstrahlung.Note that the PMT exhibits a spurious short in the signal after a few hundred nanoseconds.

80 100 120 140 160 180 200 22010

100

1000

10000 PMT voltage 7th order fit 11th order

PMT s

ignal v

oltage

(mV)

Laser Energy (µJ)0.0 1.0x10-7 2.0x10-7 3.0x10-7

0.0

0.2

0.4

0.6

0.8

1.0

norm

alized

PMT s

ignal

volta

ge (a

rb. un

its)

time (s)

5cm focal length lens: 1/e=37ns 25cm focal length lens: 1/e=2.66ns

(a) (b)

Figure 10. Temporal detection of laser-generated(λ=1054 nm,φ=12 mm FWHM, t∼500 fs) plasma fluorescence inambient air. Detection occured via a Photonis XP2020/UR PMT(1.6ns risetime) with UG11 (1mm thick)+BG39 (1mm thick)filtering (a) Temporal response of PMT for two different focallengths: f=5 cm and f=25 cm. (b) Averaged PMT signal amplitudeversus laser energy for f=25 cm case.

In Figure 10b, the peak signal output voltage of the PMT was averaged and plotted as a func-tion of laser energy. Ideally, for MPI only using 1054 nm (1.18 eV) light, ionization of O2 (at anionization potential of 12.1 eV) requires 11 photons and N2 (at an ionization potential of 15.6 eV)requires 14 photons, meaning that the oxygen will ionize first and dominate the process. If theionization process only included MPI (without any inverse Bremsstrahlung or tunneling ioniza-

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tion contribution), the slope of the data on a logarithmic plot would then be 11. While a slope of11 is shown in the figure, one might more easily assume that a lower slope of around 7 matchesthe data better. Based upon our experiences with intensity clamping in the wavefront sensor data,one expects that a contribution from tunneling ionization is present, which lowers the effetiveslope. To verify this, we take the 100 µJ to 200 µJ range of the plasma fluorescence data in Fig-ure 10b and calculate via Zemax software [39] the focal intensity of the 12 mm FWHM diameterapodized flattop beam focused by the f = 25 cm lens, which gives intensities of 3.8x1013 W/cm2

to 7.6x1013 W/cm2 (which is well below the region of intensity clamping and large nonlinear Kerreffects). The slope of the PPT curve is taken at the beginning and end of this intensity range.The slopes vary from 7.5 down to 6.8 over this range, agreeing well with the measured data inFigure 10b.

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3 Tailored and Enhanced Plasmas

With the tools in hand, the ionization control must be increased, empasizing the ability to tailor ourplasma length, lifetime, and density. We investigated this with several methods of achieving thesegoals, including sequences of multiphoton ionization (MPI) ultrashort laser pulses and secondaryheating pulses (utilizing photodetachment and cascade ionization).

3.1 Plasma Lengthening

The topic of longer plasma channels is critical to generating longer plasma lengths. Historically,researchers have tried using increased an f/# for the focal optic to extend the depth of focus of thesystem (while also deleteriously increasing the laser spot size and dropping the intensity), usingstrongly spherically aberrant focal elements to blur out the longitudinal focal region [59], usingaxicons [8, 60, 61], using segmented optics with different focal lengths at each segment [61], orusing diffractive optical elements engineered to have aspects of the previously described attempts[62]. Another straightforward method in a similar context is to simply twist the lens to induceastigmatism. The slight angle of incidence ot the lens breaks the focal symmetry shifting the fociin one axis away from the focus in the other axis, leaving the area of best circular symmetry (theso-called “circle of least confusion”) in between. As the lens angle of incidence increases from0, the “circle of least confusion” moves back towards the lens. Note that this method of plasmaextension applies to both USP lasers and nanosecond lasers (see Figure 11). The effect of this isillustrated in Figure 11, using a fused silica f=75 cm lens to focus a 2.6 J average energy laser at532 nm in a 4 ns FWHM pulsewidth. As the lens tilts, a physical optics Zemax model (without anynonlinearities or plasma effects) shows the two orthogonal foci shifting away from each other asthe average intensity centroid pulls back towar the focal lens. The measured plasma fluorscencematches reasonably at the normal incidence (0 tilt) case but begins to disagree more in amplitudeas the lens tilt increases. By the time the 10 case is reached, the plasma peak positions lookreasonable but the amplitude of the plasma fluorescence distribution further from the lens beginsto drop as the laser light is absorbed and scattered more by the early plasma. Note that the 0 casehas been modeled with a Sandia-developed nonlinear optical propagation code (see Section 5) withfavorable agreement as well. In comparing that data (see Figure 32) to this data (see Figure 11),note that the calculated electron density length far exceeds the length determined by the plasmafluorescence, which is in agreement with experimental data comparing the bright plasma emissionlength to electrical conductivity measurements [64].

USP lasers offer certain other options. In our case, with USP lasers, we have already demon-strated plasma channel length increases via the method of increasing the laser energy to the pointwhere multiple filamentation occurs (see Section 2). Other research has extended plasmas viapulse concatenation of multiple pulses of different temporal delay and focal elements [63]. Thismethod can be generalized to any multipulse method wherein each pulse has a different wavefrontcurvature. A brute force aproach to this can be achieved by beam splitting, varying the wave-front curvature of one beam via a telescope, and then recombining the beams before entry into acommon focal element. The difference in wavefront curvature on each beam causes each beam to

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750 760 770 780 790 800

0

50000

100000

150000

200000

250000

300000Pe

ak Inte

nsity (

GW/cm

^2)

Distance (mm)

Tilt=0o for 532nm

750 760 770 780 790 800

0

2

4

6

8

10

Plasm

a Fluo

rescen

ce Inte

nsity (

a.u.)

Distance (mm)

average of 4 data sets for tilt=0o at 532nm

750 760 770 780 790 800

0

10000

20000

30000

40000

50000

Peak

Intensi

ty (GW

/cm^2)

Distance (mm)

Tilt=5o for 532nm

750 760 770 780 790 800

0

2

4

6

8

10

Plasm

a Fluo

rescen

ce Inte

nsity (

a.u.)

Distance (mm)

average of 4 data sets for tilt=5o at 532nm

730 740 750 760 770

0

2000

4000

6000

8000

10000

12000

Peak

Intensi

ty (GW

/cm^2)

Distance (mm)

Tilt=10o for 532nm

730 740 750 760 770

0

2

4

6

8

Plasm

a Fluo

rescen

ce Inte

nsity (

a.u.)

Distance (mm)

average of 4 data sets for tilt=10o at 532nm

(a) (b) (c)

(d) (e) (f)

Figure 11. The effect of astigmatism upon plasma length, asillustrtated by 3 different lens tilts or lens angles of incidence.Figures (a)-(c) are calculated based upon a physical optics Ze-max model assuming an apodized flattop beam profile of 24 mmFWHM and a nominal 1 GW (4 J/4 ns) peak power at 532 nm. Fig-ures (d)-(f) are the longitudinal averages for 4 measured plasmafluorescence images for the nominal 24 mm FWHM beam witha measured 0.65 GW (2.6 J average/4 ns) peak power at 532 nm.Each image is summed transverse to the laser to account for in-creased plasma volume prior to the averaging.

focus at a slightly different location, stitching or concatenating the individual laser plasmas into alonger overall plasma length. This method can be applied by either a nanosecond or USP laser butthe plasma uniformity will be better with either ultraviolet pulses or USP pulses (based upon theionization mechanisms). The nanosecond laser demonstration for a 4 ns pulsewidth at 532 nm isshown in Figure 12. In this test, the fixed arm (arm 1) had 0.58 0.03 J, while the adjustable focusarm (arm 2) had 0.52 0.02 J, such that combined beams measured 1.13 0.05 J. With the commonf=75 cm lens at 0 tilt after recombination of the beams, arm 1 was in position 2 (with the resultingplasma about 1 cm below the upper positive HV electrode) while arm 2 was adjusted to positionsA, B, and C in order to create plasmas at 1, 2, and 3 cm below that of arm 1. No measureable delaywas present between the arms. The combined plasmas (as shown in Figure 12) were longer butshowed no siginificant improvement in continuity.

Another approach of the same wavefront curvature concept which can apply to either nanosec-ond or USP lasers is based on chromatic aberration. If multiple harmonics of a laser are to be used,the material dispersion of the lens will create different wavefront curvatures for each wavelength.These separate focal regions generate plasmas which can then concatenate together into a longer

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Arm 2Pos. C

Both ArmsPos. 2+C

Arm 1Pos. 2

Arm 2Pos. A

Both ArmsPos. 2+A

Arm 2Pos. B

Both ArmsPos. 2+B

+HV Electrode Surface

GND Electrode Surface

5.1cm

Figure 12. Plasma Florescence imaging of nanosecond pulselaser-generated plasma concatenation based upon pulse wavefrontcurvature.

plasma. As wavelengths in the visible and infrared range tend to exhibit plasma beading and dis-continuity, USP laser beam harmonics will be show better performance (see Figure 13). Note thatlonger focal length lenses and higher harmonic orders both show larger plasma separations whichmay not coalesce (i.e the overall plasma uniformity and continuity is poor). The effects can bereduced by increasing the f/# of the focal system (via an input beam diameter reduction) such thatthe depth of focus extends, resulting in extending these distinct plasma channels to the point wherethey join or coalesce.

A final approach of the wavefront curvature concept which is unique to USP lasers is based onself-focusing. If multiple pulses of different energy are generated, the self-focusing of each pulsewill generate different wavefront curvatures and hence different foci for each pulse. As before,these separate focal regions generate plasmas which can then concatenate together into a longerplasma. To test the concept, the USP laser pulse was split into two different energies for twodifferent polarizations (3 mJ for s-polarized and 9 mJ for p-polarized). The orthogonally polarizedpulses were recombined at the same time (zero delay). The resulting shadowgraph demonstratesthe concept in Figure 14. Note that when combined, the overall length is longer than the combinedindividual pulses. However, the overall length is nearly identical to the single pulse case shown inthe 11 mJ case in Figure 5. While this may at first seem discouraging, it actually is not since thesingle pulse case will break into multiple unconnected filaments at a few times the critical power.Using pulse combination methods, energy can be distributed amongst multiple pulses to avoid thisproblem, allowing longer continuous plasma channels in the process.

Note that time delays in these multiple pulse concatenation methods may impact switchingapplications. Also note that several of these methods can be used in conjunction with each otherto increase the overall effect

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1mm

1mm

0 ps

1.3 ps

2.67 ps

4.0 ps

0 ps

1.3 ps

2.67 ps

4.0 ps

Beam Direction

Beam Direction

(a)

(b)

Figure 13. USP laser-generated plasma concatenation basedupon chromatic aberration. The figures have been false coloredhorizontally to correlate to the ionizing wavelength. (a) Shad-owgraph of 1054 nm, 527 nm, and 263.5 nm USP at ∼12 mJ and∼500 fs focused with an f=7.5cm lens (b) Shadowgraphs at vari-ous probe delays with an f=5cm lens looking at the coalescence oftwo plasma channels from 1054 nm and 527 nm pulses.

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s-polarization only 1mm

p-polarization only 1mm

s and p-polarization 1mm

Figure 14. USP laser-generated plasma concatenation basedupon self-focusing. The laser pulses (λ=1054 nm,φ=12 mmFWHM, t∼500 fs) are entering from the left with 3 mJ for the s-polarized pulse and 9 mJ for the p-polarized pulse, as focused byan f=25 cm lens in ambient air.

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3.2 Plasma Lifetime Increases

For a given laser energy budget, effective plasma lifetime can be shown to increase in two ways:multiple laser pulsing in rapid succession so as to re-ionize a volume [65–67] and multiple laserpulsing with one laser providing seed electrons for another longer pulse laser to heat via inverseBremsstrahlung [68–70]. The latter effect usually involves a both large electron density and tem-perature increase, with both effects increasing the corresponding recombination time [14]. Thisapproach will be discussed under the next subsection.

The first technique usually uses USP lasers which ionize weakly and have short lifetimes suchthat the subsequent pulses are not greatly distubed by the pre-existing plasma. Having said this, thetime frame between pulses must be short enough such that the electron density does not fully decaybefore secondary laser pulses enter the region (lest the effective lifetime not be really increased).We demonstrate this via orthogonal USP pulses generated by the setup in Figure 2. Each USP has3 mJ energy each and has been focused with an f=25 cm lens. The time delay between pulses wasadjustable from the simultaneous (zero-delay) case to several nanoseconds. The plasma fluores-cence emission is then monitored via a UV-filtered PMT (as was used for the data in Figure 10).The data, shown in Figure 15, shows an increase in flourescence width and amplitude as the secondpulse is added. The additional energy increases the electron number density but also increases theionized volume, correspondingly raising the overall signal due to the greater total number of re-combining molecules. As the delay between pulses increases, the fluorescence width increases. Ata 3.8 ns delay, we begin to see the distinct fluorescence peaks of each pulse. Note that the overall1/e width of 14.9 ns exceeds the sum of the two individual widths (3.6 ns) plus the delay, implyingthat some other effects such as plasma heating by the second pulse may be occurring.

7.50E-008 1.00E-007 1.25E-007 1.50E-0070

1

2

3

4

5

PMT s

ignal

volta

ge (V

)

time (s)

one beam only: 1/e=3.6nsTwo beam time delay:

0ps: 1/e=8.0ns 47ps 93ps 207ps 2.1ns: 1/e=9.8ns 3.8ns: 1/e=14.9ns

4ns

Figure 15. Plasma lifetime enhancement by double USP pulseswith variable delay.

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3.3 Plasma Density Increases

The concept of used USP lasers with nanosecond heating beams is of growing interest in the liter-ature [68–70]. Essentially, the USP operates at a low level to provide a small seed electron distri-bution which can initiate laser-induced breakdown (LIB) in air caused by inverse Bremsstraahlungheating from a subsequent nanosecond-scale laser pulse. We tested the concept by making a USPlaser at 1054 nm and a nanosecond laser at 532 nm collinear and cotemporal (with the latter mean-ing that the USP was placed in the center of the 8 ns pulse at 532 nm). The beams are then focusedby a common f=25 cm lens. Shadowgraphy and temporal parametric fluorescence measurementswere then made (see Figure 16).

The first observation is that the USP allows air breakdown at energies far below the 59 mJ LIBthreshold required by the nanosecond beam alone. With USP electron seeding, the longer pulsecould create LIB in air with only 30 mJ, a significant energy reduction. Note that the ignition ofthe LIB process has a short incubation time, as evidenced in the two different probe delays for theshadowgraphs in Figure 16b. In addition, the process increases the plasma density and temperatureto allow for longer lifetimes (as seen in the plasma flourescence PMT data in Figure 16a). Tocompare, the USP beam alone in this configuration has a 1/e plasma lifetime pulsewidth of 8.4 nsfor 6 mJ and 27 ns for 9 mJ while the nanosecond pulse alone has a 1/e pulsewidth of 38 ns for59 mJ (not shown in the figure). The USP plus nanosecond pulse sequence shows odd behaviorin the time domain: At lower nanosecond energies, the dual pulse plasma lifetime is the same orlonger than the USP alone while at higher nanosecond laser energies, the lifetime actually reduces.This may be due to a faster rise in plasma density for the higher nanosecond energy case, leadingto a plasma which scatters, reflects, and refracts more of the heater beam laser light. This would inturn prevent efficient coupling of the latter portion of the nanosecond heater pulse and thus resultin a cooler plasma with a shorter plasma lifetime.

In addition, we should point out that dual pulse plasma using the 532 nm nanosecond heaterbeam appears to show discrete beading structure in the plasma channel. In self-breakdown laserplasmas, such beading becoming more prominent for longer wavelength light of similar duration.This is due to plasma refraction and absorption issue at these wavelengths, as based upon theplasma gradients which result from the prevalent cascade ionization. One might hope that seedinga cascade ionization process with a uniform initial electron distribution (as one sees with USP MPIplasmas) could lead to a more uniform plasma channel out of the cascade process with a lowerenergy requirement. However, as shown in the shadowgraphs in Figure 16, beading structures arebeginning to appear in the heated plasmas. This implies that USP laser-generated plasma seedingof a cascade ionization process from a nansecond laser might not lead to the most uniform plasmadistributions, which was subsequently pointed out by Polynkin et al [70] during the course of ourresearch.

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5.00E-008 1.00E-007 1.50E-0070

1

2

3

4

5

6

PMT

signa

l vol

tage

(v)

time (s)

USP: 6mJ: 1/e=8.4ns USP: 6mJ, ns<LIB: 30mJ: 1/e=30ns USP: 9mJ: 1/e=27ns USP: 9mJ, ns<LIB: 30mJ: 1/e=28ns USP: 9mJ, ns>LIB: 150mJ: 1/e=16ns

∆τ=1.74ns

1mm

∆τ=1.74ns

∆τ=1.74ns

∆τ=30ps

∆τ=30ps

∆τ=30ps

∆τ=30ps

∆τ=30ps

(a) (b) USP(6mJ)

USP(6mJ)+ns(30mJ)

USP(9mJ)

USP(9mJ)+ns(30mJ)

USP(6mJ)+ns(150mJ)

Figure 16. Dual pulse USP plus nanosecond laser plasma gen-eration. (a) Plasma lifetime enhancement by USP+ns laser, as de-tected by plasma fluorescence PMT method. (b) Shadowgraphyof resulting laser plasma for different cases, with pulses enteringfrom the left. Probe beam delays ∆τ are with respect to the time ofpeak ionization (zero-delay between the ionizing and USP lasers).

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4 Laser-Triggered Discharges

4.1 The Laser System

Based upon the size of the high voltage setup, this setup could not be located near the previouslyused laser system. As such, two other laser sources in a larger work area were considered for thelaser triggered discharge work: A developmental optical parametric chirped pulse amplification(OPCPA) USP laser at 1054 nm wavelength and a pulsewidth of 100 fs, and its pump laser source,a 5.5 J, 532 nm laser with a pulsewidth of 4 ns FWHM. Due to time constraints, the work on theshort pulse system could not be completed. However, a full system design was performed (and is tobe published) and all hardware (including the temporal compressor) are in place. In the followingsection, we will discuss the performance of the long pulse pump laser source (used in the dischargework) and the current status of the short pulse system.

The OPCPA pump laser

This commercial pump laser is specifically designed for pumping of an OPCPA system. It runs at2 Hz and generates 5.5J per pulse at 532 nm, with a 4 ns flattop temporal pulse shape. The initialpulse is sliced out from a 40 mW, 1064 nm single longitudinal wave fiber laser by a Mach-Zehnderintensity modulator driven by an arbitrary waveform generator. The pulse is then amplified to 9.5 Jat 1064 nm by a 5 mm diameter preamplifier head, a regenerative amplifier, two 12 mm diameteramplifier heads, two 19 mm diameter amplifier heads, and two 25 mm diameter amplifier heads,and then is finally frequency doubled to yield 5.5 J per pulse at 532 nm. Figure 17a shows a repre-sentative pump beam near field at the output surface of the doubling crystal. One can see that a 5thorder Gaussian function (of the form (e−x2

)n, where n is the super Gaussian coefficient) is a goodfit for the horizontal lineout (see Fig. 17b). Correspondingly, the temporal trace (not shown here)shows excellent agreement with a 4th order Gaussian fit (R2 = 0.998).

This laser source was used at an energy of 2.5-2.7 J for the laser triggered discharge workdiscussed below. Furthermore, the 532 nm output was frequency doubled to 266 nm for a small setof discharge experiments at UV wavelengths. Due to the damage threshold of our optics and theobtainable conversion efficiency, UV energies were used around 600 mJ.

The OPCPA system

The first OPCPA stage consists of two 25 mm long walk-off compensated LBO crystals that arepumped with 100 mJ of 532 nm light. Stage two has two 13 mm long walk-off corrected LBOcrystals pumped at 150 mJ of green light. Stages three and four consist of 6 mm and 9 mm thickBBO crystals with pump energies of 400 mJ and 4 J each. Due to the high pump beam energies,careful ghost analysis had to be performed in order to avoid damage on pump beam optics.

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15 20 25 30 35 40 450

50

100

150

200

250

cam

era

pixe

l cou

nt (a

rb. u

nits

)

beam size (mm)

Lineout 5th order Gaussian fit: R2=0.98251

(a) (b)

Figure 17. OPCPA pump laser performance: (a) Pump laser nearfield at the plane of the doubling crystal. (b) Horizontal line-outwas fitted with a 5th order Gaussian function and shows a FWHMof 25 mm.

The pump system is designed in such a way that the initial beam near field at the plane of thepump laser doubling crystal is spatially filtered and relay imaged onto the crystal face of all fourOPA stages to ensure good spatial beam quality. Pump and signal beam sizes have been wellmatched at the OPA crystal planes and pump intensities have been kept below 800 MW/cm2 inorder to avoid laser damage and parametric fluorescence which will ensure high temporal contrast.In addition, great care was taken to verify that signal and pump beam paths have exactly the sametime of flight, again mitigating poor temporal contrast. Furthermore, non-collinear pumping anglesare < 2 but > 0 in order to spatially separate signal (1046 nm - 1062 nm) and idler (1066 nm -1082 nm) beams after amplification while maintaing the high gain of near collinearity. Spatialseparation of the idler is required since no dichroic beamsplitters exist that have the necessaryextinction and/or rapid transition over only 4 nm bandwidth while having a damage threshold thatcan withstand our fluences. Lastly, spatial relay filters have been employed for the signal beambetween each OPA stage in order to improve beam near field quality and mitigate any residualparametric fluorescenece. Figure 18 shows a layout of the overall four stage OPCPA system,including all four OPA pump relay vacuum telescopes.

Modeling for all OPA stages was performed using a modified version of SNLO [71]. Thecode was expanded to allow up to 10th order Gaussian temporal and spatial input beam shapes aswell as an option to parametrically scan through several nonlinear crystal thicknesses. Based onthe modeling, one predicts a 4 ns FWHM temporally chirped beam with a spectral bandwidth of16 nm and an output energy of up to 1.5 J.

Temporal Compressor

For use of a short pulse laser in laser triggered discharge work, one has to consider the accumu-lation of nonlinear phase and subsequent beam break-up as the short pulse traverses through air

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MIRA 900-F (80MHz,1054nm,100fs)

VERDI

1740l/mm Stretcher

From Pump Laser

To Temporal Compressor

OPA4

OPA3

OPA2

OPA1

= beamsplitter = mirror

= OPA crystal = dichroic beamsplitter/combiner

Figure 18. Overview of the new four stage OPCPA system as itintegrates into the existing short pulse oscillator, stretcher setup.The pump laser, housed on a 4′×8′ table, is not shown here. Mas-ter oscillator and temporal stretcher are located on a 5′×12′ tablewith remaining table space for diagnostics. The actual OPA stagesare placed on the adjacent 5′ × 12′ optics table with the outputleading to the 4′×8′ temporal compressor table (not shown here).

toward the high voltage testbed. This will set a limit for the maximum laser intensity that cansafely be used. The so called “B-Integral” is a measure of nonlinear phase and its value should bekept below π/2 for safe operation.

B =2π

λn2

∫ L

0I(l)dl ≈ 2π

λn2IL≤ π

2, (20)

where λ = 1054 nm is the central wavelength of the laser, n2 = 3×10−19 W/cm2 is the nonlinearindex of air, I ≤ 1011 W/cm2 is the maximum safe intesity before beam break-up and damageoccurs, and L = 8 m is resulting distance at which the B-Integral exceeds π/2. For the gratingin question (here 42× 21 cm2 gold at 1480 l/mm), one has to find a balance between maximumbeam size (in order to support maximum power) and maximum spectral transmission (in orderto generate this power through temporal compression). Figure 19 shows such a configuration inwhich we have used a beam expander to generate a 73 mm diameter beam which can clear 14 nm(1046 nm to 1060 nm) spectral bandwidth through the compressor. At this beam area of 42 cm2, anoutput power of 4.2 TW (480 mJ/115 fs) can be supported assuming 100 GW/cm2 as a conservativelimit for filamentation in the air compressor. This can easily be generated with the available signalenergy of 1.5 J and the bandwidth limited temporal compression of≤ 150 fs. Note that commercialsystems push the intensity limit for air grating compressors to > 200 GW/cm2, which could yieldup to 10 TW operation.

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Jens SchwarzSandia National Laboratories

Power-OPCPA-1480-gold-compressor-MLD-stretcher-compensation-folded-10-10.zmxConfiguration 1 of 1

3D Layout

5/3/2012

X

Y

Z

Figure 19. Zemax model of the temporal compressor layout.The minimum transmitted wavelength of 1046 nm is depicted inblue, the central wavelength of 1053 nm is green, and the maxi-mum transmitted wavelength of 1060 nm is shown in red.

Current Status

All hardware has been assembled, all optics are in place, and all vacuum relay telescopes have beenleak checked and are operating below 150 mTorr. The signal and pump beams are aligned throughall four OPA stages and spatio-temporal overlap between pump and signal beam has been achievedat OPA stage 1. Pumping of OPA stage 1 is currently underway and initial gain has been observed.The temporal compressor has been built and was carefully aligned using a 1054 nm continuouswave alignment beam. All diagnostic cameras and energy meters are in place and their respectivedata acquisition software is functional.

To summarize, a 2 Hz OPCPA system has been designed that is capable of 4.2 TW (480 mJ at115 fs) beam power with no need for a vacuum grating compressor. Every stage in the system hasbeen designed to be insensitive to the input energy (except for stage one). All simulations representan idealized case and one should expect less performance than is modeled. This has been takeninto account when choosing the input energies for each stage. The output spectrum from eachstage has been modeled and shows no unusual behavior. System activation is underway with thefirst OPA stage being pumped right now.

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4.2 The High Voltage System

The high voltage system design quickly became highly constrained by safety and lab environmentissues. The net issues were as follows:

• Electrode Choice: The switch should be like other laser-triggered sections at Sandia.

Result: Unused electrodes from Z20 system were obtained. These reflect the designin Figure 1. However, a slight modification was made to the electrode insert on the sideopposite the entrance electrode. In SF6 LTGS, the argument is made that little beam energyreaches the opposite side such that a blank-off insert is used. Our switch anticpated the useof air (see below) which is not so absorptive. To avoid switch ablation issue impacting theperformance, we had a small exit hole drilled in this blank-off insert.

• Cleanroom: The switch design should be compatible with a cleanroom work environmentsince the lasers reside there.

Result: We will use an oil-free design to avoid organic vapors depositing films on theoptical systems. SF6 should be avoided to miminize additional hazards and complexity.This drives the design to larger high voltage stand-off distances for air and to slightly lowervoltages.

• Electrical Safety: Sandia electrical safety requires either very large (conservative) stand-offdistances for exposed high voltage or requires the high voltage to be enclosed.

Result 1: We chose to reduce the energy to less than 10 J to reduce safety issues. Theresulting HV decay time is less than 1 s, further alleviating some of the typical safety con-cerns.

Result 2: We chose to enclose the switch for safety, housing it in a roughly 3 X 4 X 7ft3 surplus Faraday shield enclosure. Additionally, engineered interlocks were required toavoid easy access to the exposed high voltage interior. As such, the Faraday enclosure has amagnteic latch on the door with a timed interlock feeding back to the commercial HV powersupply. When entry into the Faraday enclosure is necessary, the power supply is turned downand the power is turned off, allowing the small capacitor bank to quickly discharge to ground.Using an access control panel with timed relay in conjunction with the magnetic door lock,a code is entered to permit entry into the enclosure after 30s, allowing a full bleed down ofany residual stored energy in the interim. When the door is opened, a grounding stick canconnect to the HV electrode, redundantly verifying safety. Note that the HV power supplyis interlocked whenever the Faraday enclosure door is open.

• Electrode Gap: The switch design should have similar gaps to that of Z LTGS (or longer).

Result: We will have an adjustable gap separation from a few centimeters to 30 cm. Thestandard minimum gap is 5.1 cm as compared to 4.6 cm on Z LTGS.

• Ambient air: Consider ambient air gap rather than pressurized air or SF6 to mitigate com-plexity.

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Result: This simplification allows us to eliminate pressure safety hazards assocaited withestablishing a pressurized cell while allowing the basic physics to be explored. Eliminationof brittle barrier windows and lenses in close proximity to the switch volume reduces laserdamage and debris depostion issues. Finally, the lack of a pressurize vessel cylinder facili-tates diagnostics access to the switch volume. Note that, at 5390 ft which is the lab elevation,the atmospheric pressure is 83.1 kPa.

• E/p Levels: The switch should have lower switch currents and energies than Z LTGS forsafety while having similar E/p values for the switch dynamics in the trigger section.

Result: Glassman dual polarity option 150 kV DC HV power supplies were purchased.As these are oil-free and were designed for air at sea level, the applied supply voltage shouldnot really exceed about 80% of this value or 120 kV, based upon scaling to local ambientpressures. For the voltage divider used in the switch, 86.9% of the applied voltage reaches thebiased electrode in the switch. This yields up to 104 kV at the top electrode. Measurementsmade at 1 bar (100 kPa) at 20 on the breakdown voltage VBreakdown in kV in air in a uniformelectric field [72] follow the relation:

VBreakdown = 6.72√

pd +24.36(pd), (21)

for pressure p in bar and electrode gap d in cm. At 1 bar and 1 cm, the voltage is thus 31.08kV(for an electric field strength of 31.08 kV/cm and and a breakdown E/p of 31.08 kV/(cm·bar)).At our elevation, for 0.831 bar and 1 cm, the voltage from Eq. 21 is thus 26.37kV (for an elec-tric field strength of 26.37 kV/cm and and a breakdown E/p of 31.7 kV/(cm·bar)). We canachieve 104 kV over 5.1 cm which is 20.4 kV/cm or 77% of the calculated air self-breakdowncondition. This is similar to the percentage of the self-breakdown E/p used on the Z LTGS.

The end result of these constraints is a fully enclosed, interlocked oil and SF6-free system witha 100 kV DC air gap, adjustable gap separation up to 30 cm, and easy diagnostic access. Thesystem is depicted in Figure 20. The Glassman HV power supply is commonly grounded to theFaraday cage and the Facility ground. The protection resistor is a 60 MΩ network composed of30 Ohmite Super MOX resistors (5” long each, 50 MΩ, part MOX96025005FVE). Five of eachresistor are bundled in parallel, with 6 such bundles placed in series (for a nominal 32” length).Each of the 6 bundle levels is rated to 72 kV, for a total stand-off voltage rating of 432kV. Since eachresistor handles 16 W, the 30 resistor array handles up to 480 W. The worst case short circuit wouldbe 150 kV/60 MΩ=2.5 mA, for a total Ohmic power of 375 W (which is easily within the arrayrating). The charging capacitor array consist of 4 doorknob-style capacitors in series (TDK UHV-11A, 1 nF, 50 kV DC rating). Charging resistors (Ohmite Super MOX, 5” long each, 100 MΩ, partMOX96021006FVE) establish a voltage divider and set reliable voltages at each capacitor. Basedupon this, each capacitor charges to 21.7% of the power supply voltage, leading to a maximumcapacitor voltage of 26.1 kV for 120 kV at the power supply. Since E = (C ·V 2)/2, each capacitoronly stores 0.34 J, totalling 1.36 J for the 4 capacitor array. This is significantly below the 10 Jsafety value for which Sandia requires higher rigor safety precautions and permissions. The loadresistor is a ceramic-style resistor from US Resistor (15” long X 1” φ at 300 Ω, rated for 150 kV,140 W continuous, and 56 kJ). Above the load resistor is a current viewing resistor (CVR) used as

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a diagnostic to monitor the current pulse out. The CVR is from T&M Research (A-2-02 in-lineCVR, 0.01 Ω with 1ns rise time and a 16 J maximum rating).

In addition to the CVR, primary dignostics include a 16 bit CCD (which side-on images theplasma and/or discharge through the Faraday cage view port), a photodiode measuring the time-of-arrival of the ionizing laser pulse into the trigger gap, and a remote photodiode collecting plasmalight emision through the view port. A timing fiducial to the first photodiode is used to trigger aTektronix 3.5GHz, 40GS/s oscilloscope looking at all the three temporal signals (the CVR and thetwo photodiodes). Runtimes are defined by the time from the turn up in the fiducial to the timewhen the CVR similarly turns up from its baseline. Jitters are taken as the RMS deviation in 20sample runtimes.

Magnetic Latch

CVR

Load R

Cap’s + Charging R’s

Protection R’s HV Input

Switch Gap

36”X42”X87” Interior

f=75cm

CVR

Load

R

Prot

ectio

n R’

s

Entry Keypad

Glassman PS Glassman PS

Magnetic Latch

Power Supplies/Relay

Enclosure

View Port

Caps

/ Ch

argi

ng R

’s

+HV

GND View Port

Laser Entrance

Laser Entrance

(a) (b)

Figure 20. High Voltage switch stand. (a) Schematic. Note thatthe green paths represent ground. (b) Photograph.

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4.3 Nanosecond Pulse Laser-Triggered Discharges

532 nm studies

The HV switch was initially used with the 532 nm OPCPA pump source at 2.63±0.09 J to exam-ine the effects of plasma placement and length upon switch performance (5.08 cm gap at up to104 kV) with ambient air. At these energies, brightly luminous plasmas are formed with a fewten’s of nanoseconds lifetime, meaning that the air is fully singly ionized (i.e. the electron numberdensity euquals the normal number density of air molecules) and that the lifetime is of the order of(or longer than) the runtime of the switch. Thus, the only variables to switch performance shouldby plasma length, distrubtion, and placement. To investigate this, we did parametric scans of thefocal lens position (which is a fused silica f=75 cm plano-convex lens), applied high voltage, andlens tilt. Representative images of the side-on plasma fluorescence before and during discharge areshown in Figure 21. Note that the low capacitor values in the switch charging circuit make the dis-charge channels less luminous than the laser-generated plasma which initiates the discharge. Thedischarge images also show very straight guidance where the laser plasma is not luminous, againpointing to lower electron densities which are electrically conductive but not optically emissive.Erratic unguided discharges (as in the far right image of Figure 21b) tended to be near the HVthreshold for laser-triggered switching, with the near threshold condition correspondingly havinglong temporal runtimes and high jitters. Larger gaps of 10.2 cm were successfully triggered but didshow this erratic behavior due to the lower electric fields (for the available HV) being too close tothreshold conditions. As such, the emphasis remained on the 5 cm gap studies, which correlate tothe Z LTGS better anyway.

The basic summary of the parametric scans are presented in Figure 22. Regardless of lenstilt and applied HV, the switch has minimal runtimes when the laser plasma is slightly toward thepositive HV electrode from the gap center. Also, as expected, the highest voltages have the shortestruntimes and lowest jitters for all lens positions and cases. As such, we reconsider this data for thehighest voltage (104 kV) only in Figure 23. Looking at the data in Figure 23, we see that:

1. The runtime reaches a low of 6.89 ns for the 0 case, with this case being marginally betterthan the 5 case (and significantly better than the 10 case) at almost all plasma positionswithin the switch gap.

2. The jitter reaches a low of 350 ps for the 5 case, with this case being better than the 0 case(and significantly better than the 10 case) at all plasma positions within the switch gap. Thelowest jitter for the 0 case is 600 ps, underscoring that the 5 case has almost half (or more)of the jitter of the 0 case at all positions. To compare, the 6.1 MV LTGS system used on Zshows around ∼1 ns of RMS jitter for the trigger section (at only around 1 MV in 4 to 6 barof SF6) of the gap when under 20 mJ or 266 nm laser light [3]. Runtimes there are typicallyin the 10 to 20 ns range.

3. The HV laser-triggered breakdown threshold reaches a low of 27.4±1.3 kV (for a minimumelectric field of 5.4 kV/cm) for the 10 case. Note that the plasma centroid shifts down sig-nificantly with respect to the lens position, such that the two minimum voltage lens positions

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Pos. 1 5° Tilt

Pos. 1 10° Tilt

Pos. 3 0° Tilt

Pos. 1 0° Tilt

Pos. 5 10° Tilt

+HV Electrode Surface

GND Electrode Surface

Pos. 3 0° Tilt, 104kV

Pos. 3 10° Tilt, 104kV

Pos. 5 10° Tilt, 52kV

(HVthresh =49.5kV)

Reflection Region

Reflection Region

+HV Electrode Surface

GND Electrode Surface

Switch gap (d=5.08cm)

(a)

(b)

Figure 21. Plasma fluorescence imaging for laser plasmas andtriggered discharges. Position 1 for the lens placed a very lowintensity laser spark just at the entrance to the top positive HVelectrode. At higher intensities, the laser plasma pulled back intothe switch gap. Subsequent lens positions are spaced by 1 cm. (a)Laser plasmas at a few representative positions. (b) Correspondinglaser-triggered discharges.

for the 10 case correspond to plasmas in the middle of the gap rather than near the HVelectrode (see Figure 21a).

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The point is that, while jitter, runtime, and HV breakdown threshold generally correlate, thereis no absolute relation. Generally as the high voltage lowers, the switch runtime increases andthe jitter increases, leading one to assume that the jitter correlates to the runtime. To clarify theapparent discrepancy with the data in Figure 23, we hypothesize that things correlate back to theplasma distribution:

1. Referring back to Figure 11, we see that the peak plasma fluorescence is fairly consistent forall three lens tilts, being marginally higher for the 0 case. We assume that, since the 0 caseshows a slightly higher peak laser intensity in the Zemax model, the absolute electron num-ber and volume are slightly larger for this case. If there is a high electron number density oreven better if there are more electrons in a larger diameter channel, the plasma conductanceis higher, which translates into a faster streamer velocity [8] and a shorter runtime.

2. Similarly, one can see from Figure 11 that the main plasma distribution is slightly wider at thebase for the 5 case than either of the other two cases, with the 10 case having the narrowestmain peak. A longer continuous electron distribution means that the electrical discharge hasless length over which it must stochastically develop into streamer, thus reducing the jitter.

3. When considering the overall plasma distribution to outlying low level plasmas, the 10

case is clearly longer, although it has some gaps between visibly emitting plasma regions.While a discontinuous laser plasma will result in more switch jitter, it allows plasma islandhopping for a longer straight discharge length. A longer plasma channel in turn means thatthe external electric field does not have to be as high to allow electrons to bridge the gap.

To underscore this, we used the concatenation method in Figure 12 in the switch. With arm 1at position 2 (such that the laser plasma is roughly 1 cm below the upper positive HV electrode)and arm 2 adjusted to positions A, B, and C (with corresponding laser plasma positions 1, 2, and3 cm bellow the plasma from arm 1), the associated runtimes and jitters for positions A, B, and Care 10.44± 0.31 ns, 10.13±0.22 ns, and 9.54±0.39 ns. As one can see, for the lower average laserenergy of 1.13 J in the combined arms 1 and 2 (as compared to the single pulse 532 nm case at2.63 J average), the runtime is marginally longer than the best single pulse case (see Figure 23a)but the minimum dual pulse jitter is in fact the best observed at 220 ps (for the position 2 + positionB case). The dual pulse (for the position 2 + position C case) laser-triggered HV breakdown thresh-old measured 26.1 kV (for an electric field of 5.1 kV/cm), which is still better than the minimumsingle pulse threshold at less than half of the laser energy. From the laser fluorescence images inFigure 12, the dual pulse plasma is longer albeit more discontinuous than any of the single pulseplasma cases. Based on our hypothesis above, such a laser-generated plasma will lead to switchingwith a lower jitter and lower HV threshold, as observed.

Obviously, the best scenario is a long continuous dense plasma for HV switching. But in lieu ofthis capability, the work shown points to tailored switch performance via tailored laser-generatedplasma channels.

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0 1 2 3 4 51

10

100

1000

10000

runtim

e (ns)

Lens Position (cm)

52kV (10.2kV/cm), 0o lens tilt 78kV (15.4kV/cm), 0o lens tilt 104kV (20.5kV/cm), 0o lens tilt

0 1 2 3 4 51

10

100

1000

runtim

e (ns)

Lens Position (cm)

52kV (10.2kV/cm), 5o lens tilt 78kV (15.4kV/cm), 5o lens tilt 104kV (20.5kV/cm), 5o lens tilt

0 1 2 3 4 51

10

100

1000

10000

runtim

e (ns)

Lens Position (cm)

52kV (10.2kV/cm), 10o lens tilt 78kV (15.4kV/cm), 10o lens tilt 104kV (20.5kV/cm), 10o lens tilt

(a)

(b)

(c)

Figure 22. Air switch runtimes for 532 nm at various lens tiltsand voltages across the gap.

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(a)

(b)

(c)

0 1 2 3 4 50

1

2

3

4

jitter (n

s)

Lens Position (cm)

jitter 0o lens tilt jitter 5o lens tilt jitter 10o lens tilt

All at 104kV (20.5kV/cm)

0 1 2 3 4 50

1

2

3

4

jitter (n

s)

Lens Position (cm)

jitter 0o lens tilt jitter 5o lens tilt jitter 10o lens tilt

All at 104kV (20.5kV/cm)

0 1 2 3 4 5

30

40

50

Lens Position (cm)

Plate H

V thre

shold (

kV)

HVthresh 0o lens tilt HVthresh 5o lens tilt HVthresh 10o lens tilt

All at 104kV (20.5kV/cm)

Figure 23. Air switch performance for 532 nm at various lenstilts for 104 kV across the gap.

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266 nm studies

The 532 nm laser source was converted to 266 nm via a fourth harmonic crystal (KDP) designedfor high efficiency. Of the 2.6 J at 532 nm, up to 0.9 J at 266 nm was generated but the energyquickly degraded. Over the course of 10 minutes, the energy would drop to around 0.4 J and theconversion crystal would be adjusted for optimal performance again. We feel that the heat loadfrom the laser beam was shifting the crystal tuning curve and that a heater may be necessary onthe crystal for stable operation. However, we did perform some HV testing with the erratic laserpulse, which would probably have measured 0.6±0.1 J over the experimental time frame. Theplasma position (for 0 lens tilt) in the switch gap shifted significantly from the 532 nm case dueto chromatic aberration, so position correlation to the previous work would have to be done fromthe plasma flourescecne images. A sample laser plasma position in the middle of the gap is shownin Figure 24. The visible plasmas look longer and more continuous than ther 532 nm peers (at afraction of the laser energy) but do show the impact of laser energy fluctuation upon the plasmareproducibility. The corresponding switch operation showed very straight guided discharges with aruntime of 9.7±0.32 ns. Differences in runtime as compared to the single pulse 532 nm case couldbe due to the lower laser energy but could also be due to the tighter UV focus generating a narrowand less conductive plasma channel (especially at the channel ends). The low jitter is consistentwith the high plasma channel continuity.

+HV Electrode Surface

GND Electrode Surface

Switch gap (d=5.08cm)

Figure 24. Plasma fluorescence imaging for UV laser plasmasat the same lens position (left two images). Variances in the twoimages are due to laser energy fluctuations. The far right imageshows the corresponding triggered discharge.

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5 Nonlinear Electromagnetic Propagation Model

This section describes the laser ionization and propagation modeling effort associated with theLDRD. Over the course of the period of performance we developed a complete scaled modelincluding a wide variety of physical effects, a numerical scheme for implementing a viable simu-lation of the model and Matlab codes implementing the numerical physics model. We have alsodeveloped a fully distributed (via MPI) C++ version of the code running on desktop workstationsas well as super-computers such as Sandia’s RedSky cluster and have nearly completed an adaptivemesh refinement version of the code capable of simulating a completely new class of propagationproblems. Each of these codes has been fully validated against numerous papers from the literatureand available theoretical physics models. We are not aware of any codes with results available inliterature offering the full range of physical effects available in this newly developed simulationcode. Simulations of models of this type involve a host of numerical methods. Two of the fea-tures our code possesses allow us to study and understand some of the important effects of thisaspect of the simulation. Our linear propagator allows the construction of differential operatorswith arbitrary order giving us the ability to understand the effect of varying this aspect of the nu-merical model. We have discovered that broader support for the differential operators improves thefidelity of the numerical model only up to a certain point after which increasing the order actuallydegrades the performance. Also, we can vary the type of linear system solver used in the linearpropagator. In doing so we have discovered that introduction of a matrix preconditioner can affectthe computed pulse shapes meaningfully. This implies that care must be given to understand thenumerical as well as the mathematical physics in the simulation before drawing conclusions aboutphysical pulse propagation. We also learned that the numerical model of the Raman effect must beof suffiently high order in order to correctly capture the physics.

We begin with a development of a properly scaled mathematical physics model that includeslinear diffraction, group velocity dispersion, and third order dispersion, as well as instantaneousand delayed (Raman) Kerr, linear and non-linear shock, and ionization with avalanche, multi-photon ionization (MPI), multi-photon absorption (MPA) and second order recombination. Also,available are linear and non-linear shock components. Fundamental understanding of the modelequations at a basic level is crucial to successful addition of correctly modeled and scaled physicaleffects. The development beginning with Maxwell’s equations for electromagnetic wave propaga-tion is given in section 5.1. In section 5.1 we briefly review the relationship between Maxwell’sequations in free-space, introduce models of linear and non-linear refractive index for material in-teraction and arrive at a full vector description of wave-propagation in non-linear materials withequations (68) and (69). Section 5.1 gives the scalar model (which we use), defines some keyparameters of the model and specifies the method for factoring out the fast phase variation or car-rier frequency to leave a computationally tractable envelope model. In section 5.1 we discuss thetransformation of the dispersion operator from a full convolutional description to a differential ap-proach commonly used in such models. This transform introduces the concepts of group velocityand third order dispersion as approximations to the full model leaving us with a simpler equationfor pulse propagation. By introducing a temporally retarded coordinate system traveling at thegroup velocity of the pulse in section 5.1 we further simplify the requirements of the simulation(the computational grid travels with the pulse) without approximation. Section 5.1 gives the model

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description after assumptions about the slowly evolving nature of the pulse envelope. Normaliza-tion of the equations in section 5.1 by quantities related to the pulse temporal and spacial extents ina fashion similar to that found in Zozulya [73] gives a scaled code for numerical simulation. Ourmodel is different from Zozulya’s in that we include ionization. The ionization model describedin section 5.1 gives the ordinary differential equation for the time evolution of the electron densitydensity needed for the descriptions of pulse-plasma interactions. As is standard we use the Drudemodel to relate frequency, collision time and ionization avalanche rate coefficient. However, wehave discovered in practice that this model is insufficient for some measured values of the ion-ization rate giving aphysical (imaginary) collision times. Hence we allow separate specificationof these two parameters. We have seen no discussion of this problem in the modeling literature.Finally in section 5.1 we describe the complete simulation model in summary fashion with all thenecessary scaling factors, equations, and physical simulation parameters necessary to the model.

After establishing the complete model in section 5.1 we describe briefly the numerical frame-work for the simulation in section 5.2. Comparison of the model to papers by Mlejnek [74] (forinstantaneous Kerr and ionization) and Zozulya [73] (for Raman and shock) in section 5.3 showsvery good agreement. Also, in section 5.3 we compare against some available theoretical resultssuch as diffraction and dispersion lengths and non-linear critical power Kerr-diffraction balanc-ing with excellent agreement. In section 5.3 we show a comparison of the model data with anexperiment performed in the laboratory here at Sandia.

5.1 Mathematical Physics

In this section we discuss the relationship of the mathematical model to the basic physics relation-ships of wave propagation.

From Maxwell’s Equations to the Wave Equation

We begin with Maxwell’s equations in SI units.

∇×E =−∂tB Faraday’s Law (22)∇×H = J +∂tD Ampere’s Law (23)

∇ ·D = ρ Electric Gauss’ Law (24)∇ ·B = 0 Magnetic Gauss’ Law (25)

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where

E = Electric field strength (volts/meter) (26)H = Magnetic field strength (amperes/meter) (27)D = Electric flux density (coulombs/square meter) (28)B = Magnetic flux density (webers/square meter) (29)J = Current density (amperes/square meter) (30)

ρ = Electric charge density (coulombs/cubic meter) (31)

are generally functions of time and space. In order to solve the equations for a given circumstancewe need additional eqations relating the field quantities. These constitutive relations specify therelationships between field quantities that are functions of a specific material. In free space wehave

D = εoE (32)B = µoH (33)

where

εo = 8.854×10−12 F/m (34)

µo = 4π×10−7 H/m (35)

It should be noted at this point that some of the most useful papers on non-linear propagation(see for example Brabec and Krausz [75]) use Gaussian units.1 This must be taken into accountwhen comparing results and determining the values of various parameters used for computation.A complete discussion of these differences may be found in Cohen [76].

In order to develop the constitutive relationships for a dispersive material we general move tothe frequency domain

1In the gaussian unit system the field quantities are related via

∇×E =−1c

∂tB Faraday’s Law (36)

∇×H =4π

cJ +

1c

∂tD Ampere’s Law (37)

∇ ·D = 4πρ Electric Gauss’ Law (38)∇ ·B = 0 Magnetic Gauss’ Law (39)

with correspondingly different units and free-space constitutive relationships of

D = E (40)B = H (41)

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Derivation of the Wave Equation

Several assumptions are made to be able to simplify this derivation. The first is that the propagationmedia is nonmagnetic (see Eq. (33) The optical wave equation is then derived by taking the curl ofthe Faraday’s Law of Induction equation (see Eq. (22)).

∇×∇×E =−∇×∂tB (42)

By interchanging the order of the space and time derivatives on the right-hand side we get

∇×∇×E =−∂t(∇×B) (43)

The relationship between ∇×B and D is then derived using Maxwell’s equations and the assump-tions, giving

∇×H =−∂tD +J (44)∇×B = µ0∂tD +µ0J (45)

This then can be manipulated into the equations

∇×∇×E =−µ0∂2t D−µ0∂tJ (46)

∇×∇×E +µ0∂2t D +µ0∂tJ = 0 (47)

The material is allowed to be nonlinear

D = ε0E +P (48)

where P is the polarization density vector. If the polarization P is broken down into linear andnonlinear segments then

P = PL +PN L (49)

PL = χ(1)

ε0E (Where χ(1) is the linear susceptibility) (50)

n2 = 1+χ(1) (Where n is the linear index of refraction) (51)

ε = (1+χ(1))ε0 (Where ε is the permittivity) (52)

(53)

This leads to the nonlinear form of the electric flux density of

D = ε0E +P = ε0E +χ(1)

ε0E +PN L = ε0(1+χ(1))E +PN L = εE +PN L (54)

Looking at the linear segment of the nonlinear form of the electric flux density equation it shows

DL = εE (55)

This is the form of the equation for the case of an isotropic material (material that is uniform in alldirections. If this assumption is not made then ε is a dielectric tensor and the equation is written as

DL = ε ·E (56)

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For this derivation the isotropic material assumption is used. For a dispersive medium ε is fre-quency dependent. This leads to the equation

DL (ω) = ε(ω)E (ω) (57)

whereDL (t) F−→ DL (ω) and E (t) F−→ E (ω) (58)

Converting this to the time domain using the inverse Fourier transform gives

DL (t) =∫ t

−∞

ε(t− t ′)E(t ′)dt ′ (59)

Using the nonlinear material relationship and replacing D with E and P gives:

∇×∇×E +µ0∂2t

(∫ t

−∞

ε(t− t ′)E(t ′)dt ′+PN L

)+µ0∂tJ = 0 (60)

By replacing µ0 by 1ε0c2 and using the nonlinear material relationship obtained by replacing D with

E and P we may write

∇×∇×E +1

ε0c2 ∂2t

(∫ t

−∞

ε(t− t ′)E(t ′)dt ′+PN L

)+

1ε0c2 ∂tJ = 0 (61)

This is the most general form of the wave equation for nonlinear optics. There are some caseswhere this can be simplified. One that is used for this derivation is using the identity

∇×∇×E = ∇(∇ ·E )−∇2E (62)

With no other assumptions the wave equation becomes:

−∇(∇ ·E )+∇2E − 1

ε0c2 ∂2t

∫ t

−∞

ε(t− t ′)E(t ′)dt ′ =1

ε0c2 ∂2t PN L +

1ε0c2 ∂tJ (63)

Given that the relative permittivity is:εr =

ε

ε0(64)

the equation can also be written as:

−∇(∇ ·E )+∇2E − 1

c2 ∂2t

∫ t

−∞

εr(t− t ′)E(t ′)dt ′ =1

ε0c2 ∂2t PN L +

1ε0c2 ∂tJ (65)

or in terms of the refractive index as:

−∇(∇ ·E )+∇2E − 1

c2 ∂2t

∫ t

−∞

n2(t− t ′)E (t ′)dt ′ =1

ε0c2 ∂2t PN L +

1ε0c2 ∂tJ (66)

The majority of papers and books assume that ∇(∇ ·E )≈ 0. This assumption is sometimes referredto as the slowly varying amplitude approximation. Making it gives:

∇2E − 1

c2 ∂2t

∫ t

−∞

εr(t− t ′)E (t ′)dt ′ =1

ε0c2 ∂2t PN L +

1ε0c2 ∂tJ (67)

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Substituting (∂ 2z +∇2

⊥) in for ∇2 gives:

(∂ 2z +∇

2⊥)E −

1c2 ∂

2t

∫ t

−∞

εr(t− t ′)E (t ′)dt ′ =1

ε0c2 ∂2t PN L +

1ε0c2 ∂tJ (68)

If ionization is not included (J = 0) then this is the starting equation for formulations like that inBrabec (his formulation uses Gaussian units instead of SI units) [75].

The plasma current density associated with the ionized gas is:

∂tJ =e2

mρE − ε0c∂t

∑n ρnWnKnhωo

| E |2E (69)

where quantities e and m denote electron charge in Coulombs and mass in kilograms, ρ is thefree-electron number density (units of inverse cubic length) , and n is the number of gas species.

Scalar Model

Here we develop the nonlinear envelope equation found in [75]. It is a scalar model (assumes inputpolarization is not significantly altered during propagation). We begin with the following waveequation for a dispersive material (SI units).

(∂ 2z +∇

2⊥)E(r, t

′)− 1c2 ∂

2t

∫ t

−∞

dt ′εr(t− t ′)E(r, t) =1

ε0c2 (∂2t Pnl(r, t)+∂tJ ) (70)

We intend to separate the fast oscillation of the wave group from the general envelope shape. Wemay write the complete electric field as the product of a fast phase term and an envelope. Themodel we develop here will ultimately be paraxial in nature. We therefore expect the propagationand transverse directions to be handled somewhat differently in the approximations that follow.and thus write the complete electric field component of the disturbance as follows

E(r, t) = A(r⊥,z, t)ei(βoz−ωot+ψo)+ c.c. (71)

where c.c. stands for “the complementary complex conjugate term”. Ultimately we obtain a partialdifferential equation that moves a description of the pulse envelope at the initial input plane in thetransverse and temporal coordinates to a similar description at a new location in the propagationdirection z. The phase ψo is defined so that the imaginary component of the complex envelopeA(r⊥,z, t) is zero at the input plane (z = 0). The spectral centroid of the wave group (or carrierfrequency) is defined to be

ωo ≡∫

o dω ω|E(ω)|2∫∞

o dω |E(ω)|2, (72)

the wave number βo is defined as

βo ≡ Re[k(ωo)] =ωo

cno, (73)

andk(ω)≡ ω

c

√εr(ω) (74)

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As Brabec [75] points out, the wave group must be such that its basic carrier frequency does notdepend strongly on the value of the initial fast phase component. After obtaining the envelope dis-tribution at a particular z-coordinate location we may obtain the complete electric field distributionusing Eq. (71).

Similarly the induced nonlinear polarization is written as:

PNL(r, t) = B(r⊥,z, t)ei(βoz−ωot+ψo)+ c.c. (75)

Dispersion Operator

Now let us look at the linear dispersion operator

LdE(r, t)≡ 1c2 ∂

2t

∫ t

−∞

dt ′εr(t− t ′)E(r, t ′) (76)

It is useful to manipulate this term in the frequency domain. Let

f (t) =1

∫∞

−∞

dω f (ω)e−iωt (77)

f (ω) =∫

−∞

dt f (t)eiωt (78)

define the Fourier transform. The sign of the exponent in eq. (77) is important because it deter-mines the signs of the terms involving the time derivative later. If the final differential equation issolved numerically in the temporal Fourier domain then the kernel of the discrete Fourier transform(implemented via FFT or Fast Fourier Transform routines) must match the kernel of the forwardFourier transform used to transform the temporal differential operators. With some manipulationwe may write the dispersion operator as

D =−α1

2∂t +

∑m=2

βm + iαm/2m!

(i∂t)m (79)

Retarded Temporal Coordinates

We now move to a retarded time coordinate moving with respect to the z-axis at the group velocityβ−11 . We might also view this as a z-axis moving with respect to the time coordinate at the group

velocity. Let

τ = t−β1z (80)ζ = z (81)

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Use of the dispersion operator, the temporal coordinate transform and removal of the fast phaseallows us to arrive at

(2iβo∂ζ +∂2ζ−2β1∂ζ ∂τ +∇

2⊥)A+(iαoβo−

α2o

4−αoβ1∂τ +2βoD+ iαoD+2iβ1∂τD+ D2)A

=− ω2o

ε0c2

(1+

iωo

∂τ

)2

B+1

ε0c2

[e2

mρA+ iωoε0c

(1+

iωo

∂t

)∑n ρnWnKnhω0

| A |2A]

(82)

Approximations

Assuming that the change in the envelope with respect to time is small compared to the temporalvariation of the electric field value in time (the “fast phase” we factored out)

|∂τA| ωo|A| (83)

We also use the slowly varying nature of the envelope shape with respect to the direction of prop-agation, or

|∂ζ A| |βoA|. (84)

Additionally, we assume βo≈ωoβ1 and βo 1 (which means that the group velocity is near c/no).For simplicity we define

T ≡(

1+i

ωo∂τ

)(85)

then (after a good deal of work) we arrive at the following one-way2 propagation equation[(∂ζ +

αo

2− iD)A− i

βo

2ε0n2o

T B]+T−1 1

2iβo∇

2⊥A=−T−1 i

2ε0ω0noc

[e2

mρA+ iωoε0cT

∑n ρnWnKnhω0

| A |2A]

(86)

Leaving out the plasma current density terms on the right hand side of equation (86) we maywrite

∂ζ A =−αo

2A+ iDA+

i2βo

(1+

iωo

∂τ

)−1

∇2⊥A+ i

βo

2ε0n2o

(1+

iωo

∂τ

)B (87)

Eq. 87 is given as Eq. 6 in [75] (Brabec assumes the plasma current density is zero in hisderivation).

The term B on the right hand side of Eq. (87) is really a non-linear constitutive relationship forthe medium in which the wave is propagating. If the material has only the nearly instantaneousKerr non-linearity we may write

B = 2ε0n0n2|A|2A = Nnl|A|2A (88)

2The term “one-way” refers to the absence of the second order derivative with respect to the propagation direction.This eliminates the ability to describe waves reflected from discontinuities in material parameters.

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where the units of the envelope A have been normalized to the square root of intensity. Also, thecoefficients are found such that B corresponds to the non-linear portion of the polarization (seesection 5.1 or Couarion [11] page 75, equation 38). If we include the delayed Kerr or Ramanresponse we write

B = Nnl

(|A(t)|2(1−α)+α

∫∞

−∞

R(t− τ)|A(τ)|2dτ

)A(t) (89)

= Nnl

(|A(t)|2(1−α)+α

∫∞

−∞

R(τ)|A(t− τ)|2dτ

)A(t) (90)

Where we have used the fact that convolution is commutative. The function R(t) is the Ramanresponse function that represents the vibrations of excited nuclei. This function has the effectof “spreading out” the waveform or pulse. It moves energy from the “front” of the pulse to the“back” of the pulse because it is a “delayed” response (as indicated via the convolution integraldescription). A common form for the function R(t) is

R(t) =1+(ωrτr)

2

ωrτ2r

exp(−t/τr)sin(ωrt) (91)

where τr and ωr are the characteristic Raman response time and frequency respectively. These willvary between materials.

Normalization

Let’s define three physically meaningful normalization lengths

τp Characteric length of the pulse in time. (92)

lD = 2τ2p/β2 Distance proportional to dispersion length. (93)

l⊥ =√

lD/2β Transverse scale factor. (94)

Using these to define the following coordinate transforms

τ = τpτ′ (95)

ζ = lDζ′ (96)

rl⊥ = r′ (97)

where r is the transverse distance measure in a cylindrical coordinate system, we have the followingdifferential relationships

dτ = τpdτ′ (98)

dζ = lDdζ′ (99)

drl⊥ = dr′ (100)

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Note that if we set αm to zero (no loss) and choose to keep only two terms of the D operator wehave from equation (79)

D =−β2

2∂ 2

∂τ2 − iβ3

6∂ 3

∂τ3 (101)

Multiplication of equation (87) by i and substituting for D we obtain with a little algebra

i∂

∂ζA− β2

2∂ 2

∂τ2 A− iβ3

6∂ 3

∂τ3 A+1

2βo

(1+

iωo

∂τ

)−1

∇2⊥A+

βo

2ε0n2o

(1+

iωo

∂τ

)B = 0 (102)

where we have again neglected loss. We use the temporal coordinate transform to obtain

i∂

∂ζA− β2

2τ2p

∂ 2

∂τ ′2A− i

β3

6τ3p

∂ 3

∂τ ′3A+

12βo

(1+

iωoτp

∂τ ′

)−1

∇2⊥A+

βo

2ε0n2o

(1+

iωoτp

∂τ ′

)B = 0

(103)Noting the coefficient of the second order temporal derivative is really 1/lD we may write

i∂

∂ζA− 1

lD

∂ 2

∂τ ′2A− i

β3

6τ3p

∂ 3

∂τ ′3A+

12βo

(1+

iωoτp

∂τ ′

)−1

∇2⊥A+

βo

2ε0n2o

(1+

iωoτp

∂τ ′

)B = 0

(104)Next we apply the longitudinal transform and multiply by lD to obtain

i∂

∂ζ ′A− ∂ 2

∂τ ′2A− i

lDβ3

6τ3p

∂ 3

∂τ ′3A+

lD2βo

(1+

iωoτp

∂τ ′

)−1

∇2⊥A+

lDβo

2ε0n2o

(1+

iωoτp

∂τ ′

)B = 0

(105)Now we observe the coefficient of the term representing diffraction (transverse Laplacian) as thesquare of l⊥ and apply the final coordinate transform to give

i∂

∂ζ ′A− ∂ 2

∂τ ′2A− i

lDβ3

6τ3p

∂ 3

∂τ ′3A+

(1+

iωoτp

∂τ ′

)−1

∇′2⊥A+

lDβo

2ε0n2o

(1+

iωoτp

∂τ ′

)B = 0 (106)

Finally we define

ε3 =lDβ3

6τ3p=

2τ2p

β2

β3

6τ3p=

β3

3β2τp(107)

and

εω =1

ωτp(108)

and write

i∂

∂ζ ′A− ∂ 2

∂τ ′2A− iε3

∂ 3

∂τ ′3A+

(1+ iεω

∂τ ′

)−1

∇′2⊥A+

lDβo

2ε0n2o

(1+ iεω

∂τ ′

)B = 0 (109)

This equation compares directly to Zozulya [73] when using equation (88) to describe the propor-tionality of the non-linear material charge displacement (polarization).

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Adding in the ionization terms and using

T ′ ≡(

1+ iεω

∂τ ′

)(110)

we have

i∂

∂ζ ′A− ∂ 2

∂τ ′2A− iε3

∂ 3

∂τ ′3A+T ′−1

∇′2⊥A+

lDβo

2ε0n2o

T ′B

−T ′−1 lD2ε0ω0noc

[e2

mρA+ iωoε0cT ′

∑n ρnWnKnhω0

| A |2A]= 0 (111)

or

i∂

∂ζ ′A− ∂ 2

∂τ ′2A− iε3

∂ 3

∂τ ′3A+T ′−1

∇′2⊥A+

lDβo

2ε0n2o

T ′B

−T ′−1 lD2ε0ω0noc

e2

mρA− i

lD2no

∑n ρnWnKnhω0

| A |2A = 0 (112)

Let

Nplasma =lD

2ε0ω0noce2

mρ (113)

and

NMPA =lD

2no

∑n ρnWnKnhω0

| A |2(114)

and we have

i∂

∂ζ ′A− ∂ 2

∂τ ′2A− iε3

∂ 3

∂τ ′3A+T ′−1

∇′2⊥A+

lDβo

2ε0n2o

T ′B−T ′−1NplasmaA− iNMPAA = 0 (115)

For a single speciesW = σKIK = σK|A|2K (116)

with definitionsρc ≡ ε0meω

20/e2

βK = ρatKhω0σK ρ = ρat−ρn (117)

we can write

Nplasma = lDω0

2ρcnocρ = lD

k0

2ρcn2o

ρ (118)

and

NMPA =lD

2no|A|2K−2Knhω0σK(ρat−ρ) =

lD2no|A|2K−2

βK

(1− ρ

ρat

)(119)

Using the Drude model frequency dependence discussed in Couarion [11] we may write the plasmaterm as

Nplasma =−ilDk0

2ρcn2o

ω0τc

1+ω20 τ2

c(1+ iω0τc)ρ (120)

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Electron Density

Our current model for ionization isdρ

dt=

σ

n2bUi

ρ|Ei|2 + |Ei|2K βK

Khω0

(1− ρ

ρat

)−αρ

2 (121)

See equation 2 from Berge et. al. [77]. The first and second terms on the RHS represent respec-tively avalanche and multiphoton ionization respectively. The final term corresponds to radiativerecombination of electrons with the atoms or molecules.

Given that

F(ρ) =σ

n2bUi

ρ|Ei|2 + |Ei|2K βK

Khω0

(1− ρ

ρat

)−αρ

2, (122)

with definitions

ν =σ

n2bUi

, K =

⌈Ui

hω0

⌉, σk =

βK

Khω0(123)

we may write more simply

F(ρ) = νρ|Ei|2 + |Ei|2Kσk

(1− ρ

ρat

)−αρ

2. (124)

We note that under the Drude model the avalanche ionization rate coefficient can be computed as

σ =ke2τ

ωmeε0[1+(ωτ)2](125)

However, this relationship will not always hold and so we will not assume it in general.

The Jacobian is then computed as

F ′(ρ) = ν |Ei|2−σk

ρat|Ei|2K−2αρ. (126)

where τ is the electron collision time and k = 2πn/λ . Also,

me = 9.10938188×10−31 kg

ε0 = 8.85×10−12 F/m

e = 1.602×10−19 C

The fundamental physics parameters of the ionization model may therefore be chosen as

nb Refractive index of mediumρat Number density of neutral atoms at (in the absence of external field)

τ Electron collision timeσ Avalanche ionziation rate coefficientUi Energy needed to liberate an electronσk Coefficient of multiphoton ionization rateα Recombination coefficient

ρb Initial electron number density (initial condition for ODE)

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where the values of τ and K are valid only for a particular angular frequency ω0. We could havechosen σ , the cross-section for inverse Bremsstrahlung instead of the electron collision frequency.However, given σ it is possible for physical values of the parameters in equation (125) to obtaintwo, distinct, real, positive values for the electron collision time. Hence we store the electroncollision time and compute the cross-section for inverse Bremsstrahlung.

Model Summary and Fundamental Parameters

The model equations are:∂zA = (L+N)A (127)

L =− i∂ 2

∂τ ′2A+ ε3

∂ 3

∂τ ′3A+ iT ′−1

∇′2⊥A (128)

N = NPlasma +NMPA +NKerr (129)

where

T ′ ≡(

1+ iεω

∂τ ′

)(130)

NMPA =− lD2no|A|2K−2

βK

(1− ρ

ρat

)(131)

NPlasma =−T ′−1lDk0

2ρcn2o

ω0τc

1+ω20 τ2

c(1+ iω0τc)ρ (132)

NKerr = i2πn2

λlD(|A(t)|2(1−α)+α

∫∞

−∞

R(τ)|A(t− τ)|2dτ) (133)

The first term is the instantaneous Kerr and the second is the delayed Kerr (Raman)

R(t) =1+(ωrτr)

2

ωrτ2r

exp(−t/τr)sin(ωrt) (134)

dt=

σ

n2bUi

ρ|Ei|2 + |Ei|2K βK

Khω0

(1− ρ

ρat

)−αρ

2 (135)

The first term is the avalanche ionization rate, the second term is the multiphoton ionization rateand the third term is the recombination rate.

ε3 =β3

3β2τp(136)

εω =1

ωτp(137)

lD = 2τ2p/β2 (138)

For these equations β3 is the third order dispersion and β2 is the group velocity dispersion.

ρc ≡ ε0meω20/e2

βK = ρatKhω0σK ρ = ρat−ρn (139)

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We note that under the Drude model the avalanche ionization rate coefficient can be computed as

σ =ke2τ

ωmeε0[1+(ωτ)2](140)

The fundamental parameters for the entire propagation model are

nb Linear refractive index of mediumGVD Group velocity dispersionTOD Third order dispersion

n2 Nonlinear refractive indexγ ratio of instantaneous to delayed Kerr (Raman)

τR Raman time constantωR Raman frequencyρat Number density of neutral atoms (in the absence of external field)

τ Electron collision time.σ Avalanche ionziation rate coefficient

σk Coefficient of multiphoton ionization rateUi Energy needed to liberate an electronα Recombination coefficient

ρb Initial electron number density (initial condition for ODE)

5.2 Numerical Methods

The basic equation usually arrived at for non-linear optical propagation is

∂zA = (L+N)A (141)

where A is the complex field envelope, L is the linear, dispersive portion and N the non-linearportion of the propagation operator.

Split-Step

Split-step formulations are based on the idea of “operator-splitting.” That is, the operator is writtenas the sum of two operators (in this case the linear and non-linear portions of the full operator)and each is applied separately. The basis for the first and second order split-step method is oftendescribed physically (as we do here). However, there is an underlying mathematical framework

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from ordinary differential equations that depends on the commutativity of the two operators3 andyields higher order split-step methods. The first-order split-step methodology simply solves twoseparate equations in succession as shown in equation (142)

∂zA′(z+dz) = LA(z) (142)∂zA(z+dz) = NA′(z+dz) (143)

using the output of one as the input to the other. This is equivalent to propagating the pulse asingle step distance in the z-direction linearly and then applying the non-linear operator to theresult concluding the first complete step. A second-order split step method involves propagatinglinearly one-half a single z-step distance, applying the non-linear operator for the full-step distanceand then completing the full-step by propagating linearly the final half-step distance. A smallnumerical investigation showed little improvement of the second order method over the first orderalgorithm and so we use the latter in all of our simulations here. In section 5.2 we describe theCrank-Nicolson discretization and linear system solver used in our simulations. Afterwards insection 5.2 we discuss briefly the non-linear operator.

Linear

The linear operator consists of both temporal and spatial derivatives. We chose in our work toapply the temporal portion of the linear operator in the frequency domain via a common Fouriertransform method. This significantly reduces difficulties with temporal numerical dispersion. Thismeans that we advance the linear spatial portion of the operator in the spatical domain once foreach frequency in the simulation. If we have Nt samples in time then we must apply the spatialportion of the operator Nt times in order to advance a single step. Here we describe that spatialoperator.

Consider a linear partial differential equation of the form

∂zu = iq1u+ iq2∇2⊥u (144)

where for our caseq1 ≡ ω

2− ε3ω3 and q2 ≡ 1+ εωω. (145)

Note that we have used a Taylor’s series expansion on the inverse shock operator to obtain thevalue for q2. In the absense of shock q2 ≡ 1.

Define the operator δ−ndmn to be an nth order derivative in space (in our case the r direction)

with support of size m. For example

d51 =

12δ

[0, −1, 0, 1, 0] (146)

also note that we have included the grid spacing. In general the operator is inversely proportionalto the nth power of the grid spacing for and order n derivative. The actual coefficients are not

3We assume that the operators commutes sufficiently over short physical distances in the propagation direction ofthe pulse.

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specified here only their support. We will discuss later how we arrive at the coefficients. Also, wewill only concern ourselves with odd support lengths.

We will say by definition the dm0 is a support centered kronecker delta function

d50 = [0, 0, 1, 0, 0] (147)

We know that in cylindrical coordinates that

∇2⊥ ≡ ∂

2r +

1r

∂r (148)

The Crank-Nicholson (CN) approximations are

u→ 12(un+1, j +un, j

)(149)

∂zu→1δz

(un+1, j−un, j

)(150)

∂2r u→ 1

2δ 2r

dm2 ·(un+1, jm +un, jm

)(151)

1r

∂ru→1

2 jδ 2r

dm1 ·(un+1, jm +un, jm

)(152)

(153)

where the notation

un, jm = [un, j−(m−1)/2, un, j−(m−1)/2+1, . . . un, j, . . . un, j+(m−1)/2−1, un, j−(m−1)/2]T (154)

describes an m-length vector evaluated at step n in the z-coordinate direction, centered at j in ther-coordinate direction. Substituting into equation (144) gives

2δz

dm0 ·(un+1, jm−un, jm

)= iq1dm

0 ·(un+1, jm +un, jm

)+i

q2

δ 2r

dm2 ·(un+1, jm +un, jm

)+i

q2

jδ 2r

dm1 ·(un+1, jm +un, jm

)(155)

[2δz

dm0 − i

(q1dm

0 +q2

δ 2r

[dm

2 +1jdm

1

])]·un+1, jm =

[2δz

dm0 + i

(q1dm

0 +q2

δ 2r

[dm

2 +1jdm

1

])]·un, jm

(156)Given that q1 and q2 are real, if we define a row vector

pmj ≡

2δz

dm0 − i

(q1dm

0 +q2

δ 2r

[dm

2 +1jdm

1

])(157)

then we may write equation (156) as

pmj ·un+1, jm = pm

j∗ ·un, jm (158)

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Note that at j = 0 we must use L’Hospital’s rule to arrive at

pm0 ≡

2δz

dm0 − i

(q1dm

0 +2q2

δ 2r

dm2

)(159)

Equation (158) is an algebraic equation in m unknowns. We know the un, jm (they are the field val-ues at the particular frequency specified by the values of q1 and q2 as a function of radial coordinateat z-step index n) and the entries of the row vectors pm

j . The RHS of equation (158) is a scalar.The m unknowns are un+1, jm (the field values at the particular frequency specified by the values ofq1 and q2 as a function of radial coordinate at z-step index n). If we add another equation centeredat an adjacent value of j (the radial coordinate index) then we have an additional equation and anadditional unknown (two equations in six unknowns). Each time we add an adjacent equation weget an additional variable. Let’s say we add an adjacent equation until we reach a boundary. Thenew variable added by each boundary equation is known. At each boundary we add an equationwith a known additional variable (m−1)/2 times. After reaching both boundaries we have exactlythe number of equations we need to solve the system. Solving the system lets us update all theun, j to un+1, j for all j at once. This method (CN) is implicit (uses both the current and futuretime to obtain the future time) and hence requires a solve (in this case a linear solve) at each step.It is unconditionally stable and will tolerate a larger step size than explicit methods (which areonly stable if the step size is chosen according to the Courant stability criterion). However, theyrequire a global solve at each step. Also, equation (157) cannot be evaluated at index j = 0. Thisproblem results from the definition of the Laplacian operator in equation (148). Here we must useequation (159).

Practical implementation for a grid with J radial samples requires that we use equation (157)and the known current values of un to create a matrix equation

Lun+1 = L∗un = vn (160)

where L is an J× J matrix, vn is a known length J vector. The row j of L is formed from equa-tion (157). The top and bottom (m− 1)/2 rows are special as they touch the boundaries. If weapply symmetric boundary conditions at the j = 0 (r = 0 boundary) and zero boundary conditionsat the outer boundary and note that

pmj = [p−`j p−`+1

j p−`+2j . . . p0

j . . . p`−2j p`−1

j p`j] (161)

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where `= (m−1)/2 then we may form the augmented matrix

p−`0 p−`+10 . . . p0

0 . . . p`−10 p`0

p−`1 p−`+11 . . . p0

1 . . . p`−11 p`1

. . .

p−`J−2 p−`+1J−2 . . . p0

J−2 p1J−2

p−`J−1 p−`+1J−1 . . . p0

J−1

(162)

The zero boundary conditions on the right mean we don’t need to extend beyond the matrix edge.We simply truncate when the center of the pm

j reaches the edge (will of course always be at rowJ−1 when we start with zero).

Numerical Linear Algebra

Section 5.2 discussed generation of a large linear system used to propagate the pulse linearly foreach of its frequencies. If we have Nt samples this system must be solved as many times, oncefor each of the frequencies in the pulse. In order to solve a large system accurately and (in theC++ code) in a distributed fashion we use the Bi-conjugate gradient stabilized technique fromthe Krylov family. Also, as discussed in the validation section (see figure 31), application of apre-conditioner makes a difference in the accuracy of the pulse shape as it propagates. A pre-conditioner is simply a matrix that, when multiplied by both sides of the equation, moves theeigenvalues of the resulting system close to unity. Doing so creates a system that is easier to solvemore accurately. Our distributed solver comes from the Petsc library.

Nonlinear

The non-linear portion of the operator (see equation (142)) is treated as a simple interest equation.As long as the operator can be written so as to produce the product of a complex scalar and theoriginal solution at z, then the updated solution at z + dz can be obtained easily. It is simplycomputed as the exponential of this complex scalar (at every point on the grid) and the step-size dz.This is possible for all combinations of the physical effects discussed in the introduction and listedin the model summary in section 5.1 except shock and ionization. In this case another numericaltechnique would be required (a simple non-linear ODE integrator would work) to complete the

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non-linear numerical step. We are unaware of any literature results that contain both these effectssimultaneously and so for validation we use the simple exponential method and do not combinethese two physics effects. The code has been written so as to facilitate modification to include suchan ODE solver in the future.

Application of the instantaneous Kerr effect is accomplished in the common fashion describedabove. The Raman or delayed Kerr is described via a convolution integral and is implemented assuch in the code. Taylor series expansion of the convolution integral creates a simpler expressionand is used in some applications [78] but it is less accurate and we were able to observe differencesbetween the two.

Solution of the evolution equation for the electron density requires use of an ODE (OrdinaryDifferential Equation) solver. The equation itself is stiff in the sense that multiple time scales areimportant (the solution moves very quickly in the presense of the pulse and slowly after the pulsepeak has passed). Some investigation was required in order to find solvers the could accuratelyintegrate the equations. We use ode23 and ode45 in Matlab depending on the application. In theC++ code we use one of the stiff integrators from the Sundials package.

5.3 Validation and Evaluation of Models

The equation for critical power in a Gaussian beam is

Pc =λ 2

2πnon2. (163)

The simulation parameters for the figures 27 and 28 are

α = 7.0×10−13 m−3/s

βK = 3.5×10−123 m13/W7

K = 8Ui = 15.76J

τc = 1.9×10−13 s

ρat = 2.7×1025 m−3

The material parameters for the figures 29, 30, and 31 are

no = 1.45

n2 = 2.5×10−20 m2/W

GVD = 3.6×10−26 s2/mγ = 0.15

τR = 50×10−15 s

ωR = 84×10−12 radians/s

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0 2 4 6 8

1.4

1.6

1.8

2

Propagation Distance (m)

Bea

m W

aist

(m

m)

Simulation versus Theoretical Beam Waist

TheorySimulation

0 0.01 0.02 0.03 0.0425

30

35

40

45

Propagation Distance (m)

Pul

se W

idth

(ns

)

Simulation versus Theoretical Pulse Width

TheorySimulation

Figure 25. The plot on the left shows a comparison between thetheoretically expected FWHM (full width half max) measurementof a Gaussian beam and the width of the beam as a function ofpropagation distance obtained in the simulation code. The e−1-amplitude radius is 1.5mm. The center wavelength is 810nm. Theplot on the right hand side shows a comparison between the the-oretical and computed dispersion of a pulse with identical centerwavelength and radius having a 30ns FWHM intensity temporalextent. The stair-step effect is a result of coarse sampling in thesimulation. The simulation medium is air.

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0 100 200 300 4004

4.5

5

5.5

6Beam Waist versus Propagation Distance

Propagation Distance (m)

Bea

m W

aist

(m

m)

0 100 200 300 4000.5

1

1.5

2Beam Intensity versus Propagation Distance

Propagation Distance (m)

Pea

k In

tens

ity (

Nor

mal

ized

)

Figure 26. The left and right plots show beam radius and in-tensity as a function of propagation distance over approximately300m in air. The center wavelength is 1024nm. The initial e−1-amplitude radius is 6mm. The input was slightly below criticalpower for a Gaussian beam as given in equation (163). Heren2 = 4× 10−23 m2/W. Clearly the beam is confined nearly in-definitely (we actually used a little less than critical power) due tothe balancing of Kerr nonlinearity and diffraction.

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Figure 27. The left figure comes from a paper by Mlejnek [74].Mlejnek’s simulation does not include shock or Raman. It doeshave instantaneous Kerr and ionization, however. Our similuationusing the same model is shown at right. The e1-amplitude radiusis 0.2mm. The temporal FWHM intensity extent of the pulse is200fs. The center wavelength is 586nm. The focus is at 2.5cm.Other relevant simulation parameters are n2 = 4.9× 1023 m2/Wand GVD = 2.6×1029 s2/m. Excellent agreement is observed.

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Figure 28. The left figure comes from a paper by Mlejnek [74].Mlejnek’s simulation does not include shock or Raman. It doeshave instantaneous Kerr and ionization, however. Our similuationusing the same model is shown at right. The e1-amplitude radiusis 0.2mm. The temporal FWHM intensity extent of the pulse is200fs. The center wavelength is 586nm. The focus is at 2.5cm.Other relevant simulation parameters are n2 = 4.9× 1023 m2/Wand GVD = 2.6×1029 s2/m. The propagation distance is 2.3cm.Excellent agreement is observed.

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0

5

10

15

Nor

mal

ized

Axi

al In

tens

ity

(a)2 cm2.25 cm2.5 cm3 cm

(b)

−100 0 100

0

5

10

15

Time (fs)

Nor

mal

ized

Axi

al In

tens

ity(c)

−100 0 100Time (fs)

(d)

Figure 29. Calculated on-axis intenisty profile (see Zozulya [73]figure 5). The propagation is in fused silica. Each plot consid-ers the following effects (a) Non-linear shock. (b) Linear shock.(c) Both linear and non-linear shock. and (d) Both shock termsand Raman effects. The center wavelength is 800nm. The initiale1-amplitude radius is 59.45 µm. The temporal FWHM inten-sity extent of the pulse is 90fs. The input intensity is 85GW/cm2.Material parameters are described in the text.

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−100 0 100

0

2

4

6

8

10

12

14

Time (fs)

Nor

mal

ized

Axi

al In

tens

ity 2 cm2.5 cm3 cm

Figure 30. The figure on the left is take from Zozulya [73] fig-ure 3 demonstrating Raman pulse splitting effects. It shows thepeak on-axis intensity normalized by the input intensity The initialpulse parameters are as given in figure 29. Material parameters aredescribed in the text.

−100 0 100

0

2

4

6

8

10

12

14

Time (fs)

Nor

mal

ized

Axi

al In

tens

ity 2 cm2.5 cm3 cm

−100 0 100

0

2

4

6

8

10

12

14

Time (fs)

Nor

mal

ized

Axi

al In

tens

ity 2 cm2.5 cm3 cm

Figure 31. Compare Zozulya [73] figure 3. It shows the peak on-axis intensity normalized by the input intensity. No preconditionerwas used in the linear solver for the data on the right. On the lefta preconditioner was used. After only a few millimeters of prop-agation differences in the pulse envelope amplitudes are apparent.The initial pulse parameters are as given in figure 29. Parametersare described in the text.

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750 760 770 780 790 800

0

2

4

6

8

10

Plasm

a Fluo

rescen

ce Inte

nsity (

a.u.)

Distance (mm)

average of 4 data sets for tilt=0o at 532nm(a)

(b)

Figure 32. The above simulation used a (3.5J)/(4ns FWHM),12.5mm 1/e amplitude radius pulse at λ = 532nm focused byf=75cm. The approximate f/# is 30 and hence the (linear) ra-dius at the focal distance should be approximately (λ )( f/#) orabout 15 µm. Using approximately 40 samples across the 15 µmradius at focus requires 100000 radial samples over the extent ofthe 30mm (to prevent edge effects near the beginning of the simu-lation). The simulation used 256 temporal samples giving approx-imately 25 million cells in the simulation at each step. A total of3000 longitudinal steps were used to propagate 1.5m (twice the fo-cal distance) saving every tenth step. The C++ code used 15 nodes(120 processors) on the Sandia RedSky system and took approxi-mately 15 hours. The structure after focus (not shown here) showssome high frequency modes of the grid that remain undamped bythe Crank-Nicolson method. A neutral species number density of2.75×1025 m−3 was used. Saturation and second order recombi-nation were turned on in the ODE solver. Observed peak electronnumber densities were approximately 1×1023 m−3.

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6 Laser-Triggered Switch Modeling

A fully-kinetic simulation of the ZR laser-triggered gas switch (LTGS) has been developed, lever-aging expertise in this area from a team at Voss Scientific. This work involves detailed finite-difference time-domain particle-in-cell (PIC) model of the trigger section of the switch and in-clude circuit modeling of other components of the ZR machine from Marx bank through thepulse-forming line (PFL). The simulations presented here build on a recently developed modelof electro-negative gas breakdown and streamer propagation [79]. Our motivation for developinga fully electromagnetic (EM) kinetic PIC description of streamer propagation in electronegativegases arises from studies of high-current, laser-triggered gas switches, such as those used on theZR facility at Sandia National Laboratories [3, 80], where SF6 is used as the working gas. Thegoal is to develop 2D and 3D models capable of treating the switch performance through the suc-cessive stages of laser-initiation of one or more thin plasma filaments between the electrodes, thesubsequent streamer/leader propagation that provides the initial current contact between the elec-trodes, and finally the formation of a high-current conduction channel. During this final stage,self-magnetic field effects associated with individual and collective conduction channel evolutionare expected to be significant and, therefore, warrant a fully EM simulation approach.

The initiation of a plasma channel and subsequent streamer and leader formation in high pres-sure devices involves coupled kinetic processes on relatively fast time scales. These processesinclude elastic and inelastic particle collisions in the gas and ambient fields, radiation genera-tion, transport and absorption, and electrode interactions. Accurate modeling of these physicalprocesses is essential for a number of applications, including high current, laser-triggered gasswitches [2, 20, 81, 82]. We make use of an implicit particle-in-cell (PIC) simulation model of gasbreakdown leading to streamer formation in electronegative gases. Within the framework of theLarge-Scale Plasma (LSP) EM PIC code, a Monte Carlo treatment for all particle interactions isused, which includes discrete photon generation, transport, and absorption for ultra-violet (UV)and soft x-ray radiation [79].

The LTGS operation relies on a sequence of conditions in the trigger gap:

1. The electronegative gas prevents breakdown of the switch below an applied voltage con-sistent with electric field to gas number density E/N ≤ 360 Td (where the unit Townsendis 1 Td=1017 V · cm2). On ZR, the trigger section of the LTGS is designed to be operatedat a voltage that is 10% of the “pre-fire” or self-breakdown voltage. In terms of E/N, thistranslates to a reduction of ∼ 50 Td below the self-triggering value.

2. The laser ionizes a filament of plasma such that the resulting enhancement increasing theE/N to a level that avalanche and UV ionization increase the filament length.

3. The filament spans the anode-cathode (AK) gap entirely and the switch closes.

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6.1 Particle-in-Cell Monte Carlo model for kinetic Simulations of laser trig-gering

The basic MCC model, described in detail for the SF6 in [79]is an extension of the model describedin [83]. This model, including electron elastic, inelastic, and ionization processes with neutrals,was benchmarked for noble gases, such as He and Ar, in which no significant electron sink ispresent. For an electronegative gas such as SF6, electron attachment processes play a key role ininhibiting breakdown at low field values. To allow for electronegative gas modeling, an attachmentalgorithm was added to the MCC model previously developed for noble gases. We have also addedan excited state neutral channel. Relatively high energy photons created by relaxation (using 0.3-ns decay time suggested in [79]) of the excited state neutrals can then be propagated through aphoton transport model. These photons, propagating at the speed of light, generate electron-ionpairs by neutral photo-absorption. This provides a non-local source of plasma generation in thevicinity of the streamer front. For computational efficiency, a relatively simple chemistry modelfor SF6 was developed. The model can easily be extended to include additional reaction channelsand particle species beyond those described here. The evolution of six particle species is tracked:electrons, neutral (ground state) SF6, a single excited neutral state SF∗6, a single positive ion SF+

6 ,a single negative ion SF−6 , and photons. We note that there are many dissociative neutral and ionicfragments which can occur in SF6 reactions [84].

6.2 Triggering with variation of laser parameters

In this section, we discuss idealized 2D simulations of the laser triggering phase of an LTGSleading to current conduction in the trigger section. The simulations are purely electrostatic withaxisymmetric geometry shown in Figure 33. We introduce a 3 to 12 ns long, 5 to 20 mJ “laserpulse” into the simulation. The laser model assumes a Gaussian radial laser envelope:

w = w0

√1+

zzR

(164)

w0=25 µm is the beam waist, zR =πw2

is the Rayleigh range (or half the depth of focus), andλ=266 nm is the laser wavelength. In the simulations, we apply the corresponding electric fieldintensity of 40 MV/ cm over a “sech2(t)” temporal profile for 20 mJ energy and 3 ns pulse length.A multiple-photon ionization probability is calculated via the Ammosov-Delone-Krainov (ADK)model [85], yielding an evolving SF+

6 -electron plasma within the laser path. This model enables usto skip over frequency and spatial scales associated with the laser beam in the simulation. It doesnot however allow for plasma feedback on the propagation of the laser pulse. For completeness, wealso permit the SF−6 to photoionize via the same model assuming an ionization (or rather detach-ment) potential of the 1.3-eV binding energy. In both cases, when a photoionization event occurs,the secondary electron is given the energy of the ionization potential with a random direction. Thiseffect is important to the seeding of the filament since it inhibits the negative ion density while thelaser is on.

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Figure 33. Idealized representation of the LTGS trigger sectionused for the 2D simulations. The laser envelope is illustrated bythe red dashed line. The electron emission site (green) providesthe final “electrical” connection for the evolving plasma filamentat the start of the conduction phase.

Also shown in Figure 33 is a region on the cathode over which space-charged-limited (SCL)emission is enabled. This region electrically connects the evolving plasma filament to the cathode,enabling the initial conduction of current in the model. This region only begins to emit once thenormal electric field at the surface exceeds a preset threshold. We typical set this threshold to a200 kV/cm value which is generally much larger than the initialized axial field. The electron emis-sion, which typically starts once the filament is within a few millimeters of the cathode, enablesa numerically “smooth” connection of the filament to the cathode. In reality, complex surfacephysics coupled to the UV emission from the approaching filament (and possibly the earlier-timelaser pulse) result in some plasma formation along the cathode surface. The SCL electron emissionmodel attempts to crudely mimic this effect.

In the first series of runs, we model the trigger closure of ZR shot 2307 which was run atroughly half voltage (3.12 MV or 460 kV in trigger section). The SF6 gas pressure in the triggeris 24.3 psi requiring a timestep of 0.067 ps to adequately resolve the collision frequency and showsimulation convergence. These parameters give E/N = 220 Td, well below that of self-breakdown.The laser parameters we treat as nominal are 20 mJ of injected energy in 3 ns with w0=25 µ mwhich gives the axial extent of the focus 2zR = 1.5 cm. The length of the filament relative to theAK gap length is important because it determines the field enhancement. The focal plane was nearthe anode, mid-gap, and near the cathode (z = 1.15, 2.3 and 3.45 cm). The evolution of the plasmafilament for these cases is shown in Figure 34. These frames plot the electron density from 4 toroughly 13 ns. The laser reaches peak intensity at 3 ns. In the relatively high E/N, the filamentgrowth is essentially an electron-avalanche rather than a pure streamer. The ADK ionization of thenegative ions allows the electron avalanche to grow rapidly with negative ion density only severaltimes that of the electrons while the laser is on. This simulation demonstrates the initial 3 phases

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of switching, by establishment of an electrical connection between the anode and cathode after12 ns and nearly 1 MA/cm2 current density in the filament. This gives a minimum trigger runtimeof 12.5 ns for near ideal conditions, well within the 10-20 ns runtime values for LTGS switches onZ, Z20 and ZR [3].

Figure 34. Electron density contours [log10(cm3) contour lev-els] at various times for Shot 2307 simulations with laser focus atz = 1.15 cm (near anode), 2.3 cm (mid gap) , and 3.45 cm (nearcathode).

The plasma channel seen in Figure 34 increases in radius at the ends and reaches a radiusof 0.06 cm by 13 ns. The electron temperature inside and outside the evolving channel is fairlyuniform at 4 eV. The ions remain relatively cool, with an average temperature of∼0.035 eV. At theend of the simulation, the plasma density exceeds 1018 cm3 or 10% ionized.

We have also simulated ZR shot 2312 which had peak 5.7 MV voltage and 51.5 psi triggerpressure. We kept the same spatial resolution although we reduced the timestep by half due to the2x larger pressure. The laser energy was 20 mJ and was focused at the gap center or z = 2.3 cm.We use this shot to illustrate the effect the filament evolution has on the electric field in the gap.In Figure 35, we see the magnitude of the electric field alongside of the electron density from 2to 9 ns. As the filament grows in density, the electric field within is reduced and the field outsideis enhanced. This process occurs once the plasma electrons exceed roughly 1013 cm3 density.Beginning with the 186 kV/cm applied field or 200 Td, the peak field increases to that of self-breakby 4 ns (before the laser shuts off) and eventually grows to 600 kV/cm or 660 Td by 9 ns. The speedof filament axial growth increases with time as the increasing enhancement would suggest givingan average 0.5 cm/ns. The speed in the cathode and anode directions is roughly equal indicatingthat the process is dominated by UV photon ionization. As with shot 2307, this process illustratesideal triggering and gives a trigger runtime of 10 ns.

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Figure 35. For shot 2312 with 5.7 MV voltage and 51.5 psi SF6pressure, the electric field (left) and electron density (right) in thetrigger gap are plotted from 2 to 9 ns.

We have also simulated several less ideal laser trigger parameters and a higher voltage shot(see Table 1). For shot 2307, these include reducing the laser power and energy by a factor of2 and 4 (denoted 1/2 and 1/4 power). We also include the one simulation of 2312 with nominallaser parameters. We also kept the energy constant by reduced the power by increasing the pulselength to 6 and 12 ns (denoted 6 and 12 ns pulse). In order to compare simulated switch closureor runtimes with measured values, we adopt a technique similar to that used by Schwarz [86] thatdefines the runtime with the peak rate of increase in the radiation as measured by the laser sparkdetector (LSD). Because they decay at a given rate and produce a photon, our measure of UV fluxis determined by the number of excited or metastable neutrals nM that when de-excited produce of15 eV photon. In the LSP simulations, we see a similar abrupt rise or inflection in nM that directlycorresponds to gap closure by the plasma. Note for the simulations with 3 different laser focalplanes, the nM inflects up at t = 11.5 ns.

In the center plot of Fig. 36, the results are plotted from the two shots with nominal laser

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Laser parameters Energy (mJ) Pulsewidth (ns) Runtime from metastableinflections (ns)

Nominal 2307 20 3 11.51/2 power 10 3 13.31/4 power 5 3 16.86 ns pulse 20 6 14.612 ns pulse 20 12 21.7Nominal 2312 20 3 9.5

Table 1. Summary of laser parameter variation simulations.

parameters which show slightly improved runtime for the higher voltage, higher pressure shot2312 compared with the mid-gap shot 2307 simulation. Finally we vary the laser parameters aslisted in Table 1. Note the increased runtime as defined by the inflection of the radiation flux tonearly 22 ns. Except for the 12 ns laser pulse simulation, these trigger runtimes are all reasonablyin agreement with the measured runtimes of 10 to 20 ns as well as current connection times in thesimulation. Also, these simulations demonstrate how switch performance can degrade as the laserpower diminishes. A 1/10 laser power simulation was not plotted because the plasma filamentnever sufficiently evolved.

Figure 36. The number of excited SF6 atoms which relates di-rectly to radiation production. All cases are for shot 2307 exceptone curve in the center plot for 2312.

6.3 Integrated Simulation of Trigger Section of ZR

We now examine the integrated behavior of the ZR power flow from the Marx bank, though thegas switch, and into the PFL using LSP with detailed circuit modeling of the various components.Shown in Figure 37, the FDTD model of the trigger section is hooked up to the ZR circuit. Weterminated the PFL downstream with a 12 Ω resistive load. This mismatch (the termination is ata 3.3 Ohm impedance line) yields reasonable reflections of 21 ns duration that produce the so-call

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squiggle - indicative of the closure of the trigger gap and capacitive couple of the voltage in thecascade section - as well as best modeling the measured voltage in the PFL.

Figure 37. A schematic of the connection of the simulation do-main to the ZR circuit upstream to the intermediate store and Marxand downstream through the PFL. The simulation domain includesonly the trigger section. The resistances for the trigger and cascadesection are shown in upper right corner.

The agreement of the measured data for shot 2307, a purely circuit representation (“circuit”)and the integrated LSP simulation (“integrated”) voltage at the PFL are shown in Figure 38. In thistest case for both the circuit and integrated simulation, the trigger section breakdown was modeledwith a time varying resistance that decreased from 106 Ω to 0.4 Ω with 1.8 ns decay time. Thecircuit and integrated simulation voltages are indistinguishable from each other. The basic rise ofthe voltage and peak are also similar to the data. Due to the simplicity of the PFL termination,however, the width of the pulse is larger. In Figure 38b, the early capacitive oscillations of thevoltage are quite similar to the data indicating we have a similar capacitive coupling after thetrigger fires. Furthermore, we have added the signal from the LSD to the figure time-shifted 21 nsto the PFL voltage position. Note the good agreement between the rise of the LSD and the squiggle.

We now replace the resistive medium in the gap with full SF6 breakdown Monte Carlo modelfor the two shots. These simulations are fully electromagnetic and require the simulation be runover 1 µs before the laser is fired and the gas breakdown begins. These simulations give an excel-lent agreement with the programed resistance model and similar squiggle and LSD response. Weare estimating switch inductances at trigger time of 75 nH also in agreement with data.

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Figure 38. The PFL voltage for shot 2307 is plotted includingdata, circuit model and LSP model. A close up of the initial rise ofthe voltage is shown in (b) showing the capacitive ringing of theswitch cascade section after the trigger section conduction. In thesimulation, the laser fires at t = 1061 ns.

6.4 Discussion and Conclusions about the PIC Model of a Z LTGS

Detailed simulations have been presented of the ZR switch with particle-in-cell kinetic modelingof the SF6 trigger section. We have examined the sensitivity of switch closure or runtime to thetrigger laser power, duration and focal plane within the switch. We found that a strong laser pulseof order 5 to 20 mJ and 3 to 6 ns is sufficient to produce runtimes < 20 ns again in agreement withavailable data. The progression of the laser-ionized filament is roughly of the same speed in theanode and cathode directions indicative of the dominance of the molecular photoionization process.Once the plasma electrons exceed roughly 1013 cm3 density, the electric field is damped withinthe filament and field enhancement occurs at the filament edges. The ionization front acceleratesacross the gap as the enhancement increases.

We have also examined the correlation between the rapid growth in measured radiation (usingthe LSD) and trigger gap runtime. In our simulations, the deduced UV photon production ratefrom de-excitation of the SF6* increases rapidly just as significant plasma density from the fila-ment spans the trigger gap and conducts significant current. The time of this conduction and theinflection in the UV production rate correlate to within 1 ns for all simulations largely validatingthe method used to calculate runtime on ZR.

In the full electromagnetic kinetic simulations of the trigger, we have been able to follow theruntime and conduction phase close to peak power which allows us to calculate the UV photonsignal. These calculations must be followed at least 21 ns after conduction to observe the squiggleat the PFL voltage monitor however. Pushing the simulations out to this point is currently beingpursued. We have also developed a technique to transition from full kinetic operation to a 2 fluidtechnique that included radiation via a multi-group diffusion model [87]. We intend to follow theplasma evolution further in time with this technique.

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7 Conclusion

Our reseach was multi-faceted and ultimately did not achieve its goal of triggering longer gaps witha USP laser, largely due to time constraints. Multiple laser plasma diagnostic methods were devel-oped and vetted. The idea of using such techniques to clearly measure laser ionization mechanismsran afoul of two problems. For nanosecond laser pulses, the issue was that plasma opacity and re-fraction made interferometry at early times impossible. For USP laser pulses, the issue was thatmultiple convoluted ionization and nonlinear effects confuse the data interpretation and corrupt thechance of using an equivalent intensity plane measurement. Ultimately, this led to the developmentof the new in-line wavefront sensor plasma diagnostic method. Using this, we were able to validatethat, at our intensity clamped values, the PPT ionization model applied, confirming that it is in-deed valid from the low pressure 10−7 Torr region up to the higher pressure 103 Torr region used inLTGS systems. With this successful, using the same plasma diagnostic appraoches, multiple meth-ods of tailoring the laser-generated plasma length, lifetime, and density were developed and/orexplored. Length increases largely dealt with the generalized multipulse method wherein eachpulse has a different wavefront curvature and thus focuses to a diferent region, creating a concate-nated plasma string. For our work, this involved using telescopes to establish different wavefronts,using chromatic aberration in focal optics to create different wavefronts at different laser harmon-ics, and using different USP laser energy levels to create different wavefront curvatures from thedifferences in self-focusing. Simple lens tilt to induce astigmatism was also explored as a methodto redistribute laser energy longitudinally. Plasma lifetime was enhanced via multiple USP laserpulses re-ionizing a region of interest within a close timeframe. Plasma lifetime and density wereboth shown to increase when using a USP laser to seed LIB from a nanosecond laser pulse, but theplasma continuity seems to suffer.

The implementation of the HV switch work was heavily impacted by safety considerations,which wound up reducing the level of HV and stored energy as well as pushing us toward a bulkyFaraday enclosure. This apparatus size in turn drove us from using the operational USP lasersystem (used for the ionization studies) to building up a developmental system where space wasavailable for the HV system. Unfortunately, although the work on the new OPCPA is nearly com-plete, the laser did not yield USP operation by the completion of the project. As such, nanosecondLTGS testing looking at the impact of the plasma distribution upon performance was conducted.The results were that various aspects of switch performance (runtime, jitter, and HV breakdownthreshold), while loosely coupled, can be observed to be adjustable depending on the plasma chan-nel parameters. Multiple laser pulse concatenated plasma channels showed the best performancewith respect to jitter and HV breakdown threshold, which (although implemented in the nansecondlaser pulse regime) was at the heart of the USP LTGS concept.

Finally, modeling was broken down to look at laser ionization and switch performance inde-pendently. Both efforts benchmark well to reality and offer new tools going forward for futurework.

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DISTRIBUTION:

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